# First power of 2 that has leading decimal digits of 12

First power of 2 that has leading decimal digits of 12
You are encouraged to solve this task according to the task description, using any language you may know.

(This task is taken from a   Project Euler   problem.)

(All numbers herein are expressed in base ten.)

27   =   128   and   7   is the first power of   2   whose leading decimal digits are   12.

The next power of   2   whose leading decimal digits are   12   is   80,
280   =   1208925819614629174706176.

Define     p(L,n)     to be the nth-smallest value of   j   such that the base ten representation of   2j   begins with the digits of   L .

```    So   p(12, 1) =  7    and
p(12, 2) = 80
```

You are also given that:

```         p(123, 45)   =   12710
```

•   find:
•   p(12, 1)
•   p(12, 2)
•   p(123, 45)
•   p(123, 12345)
•   p(123, 678910)
•   display the results here, on this page.

## 11l

Translation of: Python
```F p(=l, n, pwr = 2)
l = Int(abs(l))
V digitcount = floor(log(l, 10))
V log10pwr = log(pwr, 10)
V (raised, found) = (-1, 0)
L found < n
raised++
V firstdigits = floor(10 ^ (fract(log10pwr * raised) + digitcount))
I firstdigits == l
found++
R raised

L(l, n) [(12, 1), (12, 2), (123, 45), (123, 12345), (123, 678910)]
print(‘p(’l‘, ’n‘) = ’p(l, n))```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12710
p(123, 12345) = 3510491
p(123, 678910) = 193060223
```

## ALGOL 68

As wih the Go second sample, computes approximate integer values for the powers of 2.
Requires LONG INT to be at least 64 bits (as in Algol 68G).

```# find values of p( L, n ) where p( L, n ) is the nth-smallest j such that          #
#      the decimal representation of 2^j starts with the digits of L                #
BEGIN
# returns a string representation of n with commas                              #
PROC commatise = ( LONG INT n )STRING:
BEGIN
STRING result      := "";
STRING unformatted  = whole( n, 0 );
INT    ch count    := 0;
FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
IF   ch count <= 2 THEN ch count +:= 1
ELSE                    ch count  := 1; "," +=: result
FI;
unformatted[ c ] +=: result
OD;
result
END # commatise # ;
# returns p( prefix, occurance )                                                 #
PROC p = ( INT prefix, INT occurance )LONG INT:
BEGIN
LONG INT quarter long max int = long max int OVER 4;
LONG INT p2                  := 1;
INT      count               := 0;
INT      power               := 0;
WHILE count < occurance DO
power       +:= 1;
p2          +:= p2;
LONG INT pre := p2;
WHILE pre > prefix DO
pre OVERAB 10
OD;
IF pre = prefix THEN
count +:= 1
FI;
IF p2 > quarter long max int THEN
p2 OVERAB 10 000
FI
OD;
power
END # p # ;
# prints p( prefix, occurance )                                                 #
PROC print p = ( INT prefix, INT occurance )VOID:
print( ( "p(", whole( prefix, 0 ), ", ", whole( occurance, 0 ), ") = ", commatise( p( prefix, occurance ) ), newline ) );
# task test cases                                                               #
print p(  12,      1 );
print p(  12,      2 );
print p( 123,     45 );
print p( 123,  12345 );
print p( 123, 678910 )
END```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12,710
p(123, 12345) = 3,510,491
p(123, 678910) = 193,060,223
```

## BASIC256

```global FAC
FAC = 0.30102999566398119521373889472449302677

print p(12, 1)
print p(12, 2)
print p(123, 45)
print p(123, 12345)
print p(123, 678910)
end

function p(L, n)
cont = 0 : j = 0
LS = string(L)
while cont < n
j += 1
x = FAC * j
if x < length(LS) then continue while
y = 10^(x-int(x))
y *= 10^length(LS)
digits = string(y)
if left(digits,length(LS)) = LS then cont += 1
end while
return j
end function```
Output:
`Same as FreeBASIC entry.`

## C

Translation of: Java
```#include <math.h>
#include <stdio.h>

int p(int l, int n) {
int test = 0;
double logv = log(2.0) / log(10.0);
int factor = 1;
int loop = l;
while (loop > 10) {
factor *= 10;
loop /= 10;
}
while (n > 0) {
int val;

test++;
val = (int)(factor * pow(10.0, fmod(test * logv, 1)));
if (val == l) {
n--;
}
}
return test;
}

void runTest(int l, int n) {
printf("p(%d, %d) = %d\n", l, n, p(l, n));
}

int main() {
runTest(12, 1);
runTest(12, 2);
runTest(123, 45);
runTest(123, 12345);
runTest(123, 678910);

return 0;
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12710
p(123, 12345) = 3510491
p(123, 678910) = 193060223```

## C++

Translation of: Java
Translation of: Go
Translation of: Pascal
```// a mini chrestomathy solution

#include <string>
#include <chrono>
#include <cmath>
#include <locale>

using namespace std;
using namespace chrono;

// translated from java example
unsigned int js(int l, int n) {
unsigned int res = 0, f = 1;
double lf = log(2) / log(10), ip;
for (int i = l; i > 10; i /= 10) f *= 10;
while (n > 0)
if ((int)(f * pow(10, modf(++res * lf, &ip))) == l) n--;
return res;
}

// translated from go integer example (a.k.a. go translation of pascal alternative example)
unsigned int gi(int ld, int n) {
string Ls = to_string(ld);
unsigned int res = 0, count = 0; unsigned long long f = 1;
for (int i = 1; i <= 18 - Ls.length(); i++) f *= 10;
const unsigned long long ten18 = 1e18; unsigned long long probe = 1;
do {
probe <<= 1; res++; if (probe >= ten18) {
do {
if (probe >= ten18) probe /= 10;
if (probe / f == ld) if (++count >= n) { count--; break; }
probe <<= 1; res++;
} while (1);
}
string ps = to_string(probe);
if (ps.substr(0, min(Ls.length(), ps.length())) == Ls) if (++count >= n) break;
} while (1);
return res;
}

// translated from pascal alternative example
unsigned int pa(int ld, int n) {
const double L_float64 = pow(2, 64);
const unsigned long long Log10_2_64 = (unsigned long long)(L_float64 * log(2) / log(10));
double Log10Num; unsigned long long LmtUpper, LmtLower, Frac64;
int res = 0, dgts = 1, cnt;
for (int i = ld; i >= 10; i /= 10) dgts *= 10;
Log10Num = log((ld + 1.0) / dgts) / log(10);
// '316' was a limit
if (Log10Num >= 0.5) {
LmtUpper = (ld + 1.0) / dgts < 10.0 ? (unsigned long long)(Log10Num * (L_float64 * 0.5)) * 2 + (unsigned long long)(Log10Num * 2) : 0;
Log10Num = log((double)ld / dgts) / log(10);
LmtLower = (unsigned long long)(Log10Num * (L_float64 * 0.5)) * 2 + (unsigned long long)(Log10Num * 2);
} else {
LmtUpper = (unsigned long long)(Log10Num * L_float64);
LmtLower = (unsigned long long)(log((double)ld / dgts) / log(10) * L_float64);
}
cnt = 0; Frac64 = 0; if (LmtUpper != 0) {
do {
res++; Frac64 += Log10_2_64;
if ((Frac64 >= LmtLower) & (Frac64 < LmtUpper))
if (++cnt >= n) break;
} while (1);
} else { // '999..'
do {
res++; Frac64 += Log10_2_64;
if (Frac64 >= LmtLower) if (++cnt >= n) break;
} while (1);
};
return res;
}

int params[] = { 12, 1, 12, 2, 123, 45, 123, 12345, 123, 678910, 99, 1 };

void doOne(string name, unsigned int (*func)(int a, int b)) {
printf("%s version:\n", name.c_str());
auto start = steady_clock::now();
for (int i = 0; i < size(params); i += 2)
printf("p(%3d, %6d) = %'11u\n", params[i], params[i + 1], func(params[i], params[i + 1]));
printf("Took %f seconds\n\n", duration<double>(steady_clock::now() - start).count());
}

int main() {
setlocale(LC_ALL, "");
doOne("java simple", js);
doOne("go integer", gi);
doOne("pascal alternative", pa);
}
```
Output:
Exeution times from Tio.run, language
C++ (clang) or C++ (gcc), compiler flags: -O3 -std=c++17
```java simple version:
p( 12,      1) =           7
p( 12,      2) =          80
p(123,     45) =      12,710
p(123,  12345) =   3,510,491
p(123, 678910) = 193,060,223
p( 99,      1) =          93
Took 11.350696 seconds

go integer version:
p( 12,      1) =           7
p( 12,      2) =          80
p(123,     45) =      12,710
p(123,  12345) =   3,510,491
p(123, 678910) = 193,060,223
p( 99,      1) =          93
Took 1.787658 seconds

pascal alternative version:
p( 12,      1) =           7
p( 12,      2) =          80
p(123,     45) =      12,710
p(123,  12345) =   3,510,491
p(123, 678910) = 193,060,223
p( 99,      1) =          93
Took 0.299815 seconds

```
Execution times on a core i7-7700 @ 3.6Ghz, g++, compiler flags: -O3 -std=c++17
```java simple version:
...
Took 15.060563 seconds

go integer version:
...
Took 1.268436 seconds

pascal alternative version:
...
Took 0.082407 seconds```

## C#

Translation of: C++
```// a mini chrestomathy solution

using System;

class Program {

// translated from java example
static long js(int l, int n) {
long res = 0, f = 1;
double lf = Math.Log10(2);
for (int i = l; i > 10; i /= 10) f *= 10;
while (n > 0)
if ((int)(f * Math.Pow(10, ++res * lf % 1)) == l) n--;
return res;
}

// translated from go integer example (a.k.a. go translation of pascal alternative example)
static long gi(int ld, int n) {
string Ls = ld.ToString();
long res = 0, count = 0, f = 1;
for (int i = 1; i <= 18 - Ls.Length; i++) f *= 10;
const long ten18 = (long)1e18; long probe = 1;
do {
probe <<= 1; res++; if (probe >= ten18)
do {
if (probe >= ten18) probe /= 10;
if (probe / f == ld)
if (++count >= n) { count--; break; }
probe <<= 1; res++;
} while (true);
string ps = probe.ToString();
if (ps.Substring(0, Math.Min(Ls.Length, ps.Length)) == Ls)
if (++count >= n) break;
} while (true);
return res;
}

// translated from pascal alternative example
static long pa(int ld, int n) {
double L_float64 = Math.Pow(2, 64);
ulong Log10_2_64 = (ulong)(L_float64 * Math.Log10(2));
double Log10Num; ulong LmtUpper, LmtLower, Frac64;
long res = 0, dgts = 1, cnt;
for (int i = ld; i >= 10; i /= 10) dgts *= 10;
Log10Num = Math.Log10((ld + 1.0) / dgts);
// '316' was a limit
if (Log10Num >= 0.5) {
LmtUpper = (ld + 1.0) / dgts < 10.0 ? (ulong)(Log10Num * (L_float64 * 0.5)) * 2 + (ulong)(Log10Num * 2) : 0;
Log10Num = Math.Log10((double)ld / dgts);
LmtLower = (ulong)(Log10Num * (L_float64 * 0.5)) * 2 + (ulong)(Log10Num * 2);
} else {
LmtUpper = (ulong)(Log10Num * L_float64);
LmtLower = (ulong)(Math.Log10((double)ld / dgts) * L_float64);
}
cnt = 0; Frac64 = 0; if (LmtUpper != 0)
do {
res++; Frac64 += Log10_2_64;
if ((Frac64 >= LmtLower) & (Frac64 < LmtUpper))
if (++cnt >= n) break;
} while (true);
else // '999..'
do {
res++; Frac64 += Log10_2_64;
if (Frac64 >= LmtLower) if (++cnt >= n) break;
} while (true);
return res;
}

static int[] values = new int[] { 12, 1, 12, 2, 123, 45, 123, 12345, 123, 678910, 99, 1 };

static void doOne(string name, Func<int, int, long> fun) {
Console.WriteLine("{0} version:", name);
var start = DateTime.Now;
for (int i = 0; i < values.Length; i += 2)
Console.WriteLine("p({0,3}, {1,6}) = {2,11:n0}", values[i], values[i + 1], fun(values[i], values[i + 1]));
Console.WriteLine("Took {0} seconds\n", DateTime.Now - start);
}

static void Main() {
doOne("java simple", js);
doOne("go integer", gi);
doOne("pascal alternative", pa);
}
}
```
Output:

Results on the core i7-7700 @ 3.6Ghz.

```java simple version:
p( 12,      1) =           7
p( 12,      2) =          80
p(123,     45) =      12,710
p(123,  12345) =   3,510,491
p(123, 678910) = 193,060,223
p( 99,      1) =          93
Took 00:00:09.5992122 seconds

go integer version:
p( 12,      1) =           7
p( 12,      2) =          80
p(123,     45) =      12,710
p(123,  12345) =   3,510,491
p(123, 678910) = 193,060,223
p( 99,      1) =          93
Took 00:00:01.5335699 seconds

pascal alternative version:
p( 12,      1) =           7
p( 12,      2) =          80
p(123,     45) =      12,710
p(123,  12345) =   3,510,491
p(123, 678910) = 193,060,223
p( 99,      1) =          93
Took 00:00:00.1391226 seconds```

## D

Translation of: C
```import std.math;
import std.stdio;

int p(int l, int n) {
int test = 0;
double logv = log(2.0) / log(10.0);
int factor = 1;
int loop = l;
while (loop > 10) {
factor *= 10;
loop /= 10;
}
while (n > 0) {
int val;

test++;
val = cast(int)(factor * pow(10.0, fmod(test * logv, 1)));
if (val == l) {
n--;
}
}
return test;
}

void runTest(int l, int n) {
writefln("p(%d, %d) = %d", l, n, p(l, n));
}

void main() {
runTest(12, 1);
runTest(12, 2);
runTest(123, 45);
runTest(123, 12345);
runTest(123, 678910);
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12710
p(123, 12345) = 3510491
p(123, 678910) = 193060223```

See Pascal.

## F#

```// First power of 2 that has specified leading decimal digits. Nigel Galloway: March 14th., 2021
let fG n g=let fN l=let l=10.0**(l-floor l) in l>=n && l<g in let f=log10 2.0 in seq{1..0x0FFFFFFF}|>Seq.filter(float>>(*)f>>fN)
printfn "p(23,1)->%d"       (int(Seq.item      0 (fG 2.3 2.4)))
printfn "p(99,1)->%d"       (int(Seq.item      0 (fG 9.9 10.0)))
printfn "p(12,1)->%d"       (int(Seq.item      0 (fG 1.2 1.3)))
printfn "p(12,2)->%d"       (int(Seq.item      1 (fG 1.2 1.3)))
printfn "p(123,45)->%d"     (int(Seq.item     44 (fG 1.23 1.24)))
printfn "p(123,12345)->%d"  (int(Seq.item  12344 (fG 1.23 1.24)))
printfn "p(123,678910)->%d" (int(Seq.item 678909 (fG 1.23 1.24)))
```
Output:
```p(23,1)->61
p(99,1)->93
p(12,1)->7
p(12,2)->80
p(123,45)->12710
p(123,12345)->3510491
p(123,678910)->193060223
Real: 00:00:10.140
```

## Factor

A translation of the first Pascal example:

Translation of: Pascal
Works with: Factor version 0.99 2019-10-06
```USING: formatting fry generalizations kernel literals math
math.functions math.parser sequences tools.time ;

CONSTANT: ld10 \$[ 2 log 10 log / ]

: p ( L n -- m )
swap [ 0 0 ]
[ '[ over _ >= ] ]
[ [ log10 >integer 10^ ] keep ] tri*
'[
1 + dup ld10 * dup >integer - 10 log * e^ _ * truncate
_ number= [ [ 1 + ] dip ] when
] until nip ;

[
12 1
12 2
123 45
123 12345
123 678910
[ 2dup p "%d %d p = %d\n" printf ] 2 5 mnapply
] time
```
Output:
```12 1 p = 7
12 2 p = 80
123 45 p = 12710
123 12345 p = 3510491
123 678910 p = 193060223
Running time: 44.208249282 seconds
```

## FreeBASIC

```#define FAC 0.30102999566398119521373889472449302677

function p( L as uinteger, n as uinteger ) as uinteger
dim as uinteger count, j = 0
dim as double x, y
dim as string digits, LS = str(L)
while count < n
j+=1
x = FAC * j
if x < len(LS) then continue while
y = 10^(x-int(x))
y *= 10^len(LS)
digits  = str(y)
if left(digits,len(LS)) = LS then count += 1
wend
return j
end function

print p(12, 1)
print p(12, 2)
print p(123, 45)
print p(123, 12345)
print p(123, 678910)
```
Output:
```7
80
12710
3510491
193060223```

## Go

Translation of: Pascal
```package main

import (
"fmt"
"math"
"time"
)

const ld10 = math.Ln2 / math.Ln10

func commatize(n uint64) string {
s := fmt.Sprintf("%d", n)
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}

func p(L, n uint64) uint64 {
i := L
digits := uint64(1)
for i >= 10 {
digits *= 10
i /= 10
}
count := uint64(0)
for i = 0; count < n; i++ {
e := math.Exp(math.Ln10 * math.Mod(float64(i)*ld10, 1))
if uint64(math.Trunc(e*float64(digits))) == L {
count++
}
}
return i - 1
}

func main() {
start := time.Now()
params := []uint64{{12, 1}, {12, 2}, {123, 45}, {123, 12345}, {123, 678910}}
for _, param := range params {
fmt.Printf("p(%d, %d) = %s\n", param, param, commatize(p(param, param)))
}
fmt.Printf("\nTook %s\n", time.Since(start))
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12,710
p(123, 12345) = 3,510,491
p(123, 678910) = 193,060,223

Took 38.015225244s
```

or, translating the alternative Pascal version as well, for good measure:

```package main

import (
"fmt"
"strconv"
"time"
)

func p(L, n uint64) uint64 {
Ls := strconv.FormatUint(L, 10)
digits := uint64(1)
for d := 1; d <= 18-len(Ls); d++ {
digits *= 10
}
const ten18 uint64 = 1e18
var count, i, probe uint64 = 0, 0, 1
for {
probe += probe
i++
if probe >= ten18 {
for {
if probe >= ten18 {
probe /= 10
}
if probe/digits == L {
count++
if count >= n {
count--
break
}
}
probe += probe
i++
}
}
ps := strconv.FormatUint(probe, 10)
le := len(Ls)
if le > len(ps) {
le = len(ps)
}
if ps[0:le] == Ls {
count++
if count >= n {
break
}
}
}
return i
}

func commatize(n uint64) string {
s := fmt.Sprintf("%d", n)
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}

func main() {
start := time.Now()
params := []uint64{{12, 1}, {12, 2}, {123, 45}, {123, 12345}, {123, 678910}}
for _, param := range params {
fmt.Printf("p(%d, %d) = %s\n", param, param, commatize(p(param, param)))
}
fmt.Printf("\nTook %s\n", time.Since(start))
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12,710
p(123, 12345) = 3,510,491
p(123, 678910) = 193,060,223

Took 1.422321658s
```

Translation of: Python
```import           Control.Monad (guard)
import           Text.Printf   (printf)

p :: Int -> Int -> Int
p l n = calc !! pred n
where
digitCount = floor \$ logBase 10 (fromIntegral l :: Float)
log10pwr   = logBase 10 2
calc = do
raised <- [-1 ..]
let firstDigits = floor \$ 10 ** (snd (properFraction \$ log10pwr * realToFrac raised)
+ realToFrac digitCount)
guard (firstDigits == l)
[raised]

main :: IO ()
main = mapM_ (\(l, n) -> printf "p(%d, %d) = %d\n" l n (p l n))
[(12, 1), (12, 2), (123, 45), (123, 12345), (123, 678910)]
```

Which, desugaring a little from Control.Monad (guard) and the do syntax, could also be rewritten as:

```import Text.Printf (printf)

p :: Int -> Int -> Int
p l n = ([-1 ..] >>= f) !! pred n
where
digitCount =
floor \$
logBase 10 (fromIntegral l :: Float)
log10pwr = logBase 10 2
f raised = [ds | l == ds]
where
ds =
floor \$
10
** ( snd
( properFraction \$
log10pwr * realToFrac raised
)
+ realToFrac digitCount
)

main :: IO ()
main =
mapM_
(\(l, n) -> printf "p(%d, %d) = %d\n" l n (p l n))
[ (12, 1),
(12, 2),
(123, 45),
(123, 12345),
(123, 678910)
]
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12710
p(123, 12345) = 3510491
p(123, 678910) = 193060223```

## J

Modeled on the python version. Depending on the part of speech defined, the variable names x, y, u, v, m, and n can have special meaning within explicit code. They are defined by the arguments. In the fonts I usually use, lower case "l" is troublesome as well.

```p=: adverb define
:
el =. x
en =. y
pwr =. m
el =. <. | el
digitcount =. <. 10 ^. el
log10pwr =. 10 ^. pwr
'raised found' =. _1 0
while. found < en do.
raised =. >: raised
firstdigits =. (<.!.0) 10^digitcount + 1 | log10pwr * raised
found =. found + firstdigits = el
end.
raised
)
```
```   12 12 123 123 123 (,. ,. (2 p)&>) 1 2 45 12345 678910
12      1         7
12      2        80
123     45     12710
123  12345   3510491
123 678910 193060223
```

## Java

```public class FirstPowerOfTwo {

public static void main(String[] args) {
runTest(12, 1);
runTest(12, 2);
runTest(123, 45);
runTest(123, 12345);
runTest(123, 678910);
}

private static void runTest(int l, int n) {
System.out.printf("p(%d, %d) = %,d%n", l, n, p(l, n));
}

public static int p(int l, int n) {
int test = 0;
double log = Math.log(2) / Math.log(10);
int factor = 1;
int loop = l;
while ( loop > 10 ) {
factor *= 10;
loop /= 10;
}
while ( n > 0) {
test++;
int val = (int) (factor * Math.pow(10, test * log % 1));
if ( val == l ) {
n--;
}
}
return test;
}

}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12,710
p(123, 12345) = 3,510,491
p(123, 678910) = 193,060,223
```

## jq

This solution uses unbounded-precision integers represented as arrays of decimal digits beginning with the least significant digit, e.g. [1,2,3] represents 321.

To speed up the computation, a variable limit on the number of significant digits is used. Details can be seen in the implementation of p/2; for example, for p(123, 678910) the number of significant digits starts off at (10 + 30) = 40 and gradually tapers off to 10.

(Starting with (8+28) and tapering off to 8 is insufficient.)

```def normalize_base(\$base):
def n:
if length == 1 and . < \$base then .
else . % \$base,
((. / \$base|floor) as \$carry
|.[1:]
| . += \$carry
| n )
end;
n;

def integers_as_arrays_times(\$n; \$base):
map(. * \$n)
| [normalize_base(\$base)];

def p(\$L; \$n):
# @assert(\$L > 0 and \$n > 0)
(\$L|tostring|explode|reverse|map(. - 48)) as \$digits   # "0" is 48
| (\$digits|length) as \$ndigits
| (2*(2+(\$n|log|floor))) as \$extra
| { m: \$n,
i: 0,
keep: ( 2*(2+\$ndigits) + \$extra),
p:  # reverse-array representation of 1
}
| until(.m == 0;
.i += 1
| .p |= integers_as_arrays_times(2; 10)
| .p = .p[ -(.keep) :]
| if (.p[-\$ndigits:]) == \$digits
then
| .m += -1  | .keep -= (\$extra/\$n)
else . end )
| .i;

[[12, 1], [12, 2], [123, 45], [123, 12345], [123, 678910]][]
| "With L = \(.) and n = \(.), p(L, n) = \( p(.; .))"```
Output:

As for Julia.

## Julia

```function p(L, n)
@assert(L > 0 && n > 0)
places, logof2, nfound = trunc(log(10, L)), log(10, 2), 0
for i in 1:typemax(Int)
if L == trunc(10^(((i * logof2) % 1) + places)) && (nfound += 1) == n
return i
end
end
end

for (L, n) in [(12, 1), (12, 2), (123, 45), (123, 12345), (123, 678910)]
println("With L = \$L and n = \$n, p(L, n) = ", p(L, n))
end
```
Output:
```With L = 12 and n = 1, p(L, n) = 7
With L = 12 and n = 2, p(L, n) = 80
With L = 123 and n = 45, p(L, n) = 12710
With L = 123 and n = 12345, p(L, n) = 3510491
With L = 123 and n = 678910, p(L, n) = 193060223
```

### Faster version

Translation of: Go
```using Formatting, BenchmarkTools

function p(L, n)
@assert(L > 0 && n > 0)
Ls, ten18, nfound, i, probe = string(L), 10^18, 0, 0, 1
maxdigits = 10^(18 - ndigits(L))
while true
probe += probe
i += 1
if probe >= ten18
while true
(probe >= ten18) && (probe ÷= 10)
if probe ÷ maxdigits == L
if (nfound += 1) >= n
nfound -= 1
break
end
end
probe += probe
i += 1
end
end
ps = string(probe)
len = min(length(Ls), length(ps))
if ps[1:len] == Ls && (nfound += 1) >= n
break
end
end
return i
end

function testpLn(verbose)
for (L, n) in [(12, 1), (12, 2), (123, 45), (123, 12345), (123, 678910)]
i = p(L, n)
verbose && println("With L = \$L and n = \$n, p(L, n) = ", format(i, commas=true))
end
end

testpLn(true)
@btime testpLn(false)
```
Output:
```With L = 12 and n = 1, p(L, n) = 7
With L = 12 and n = 2, p(L, n) = 80
With L = 123 and n = 45, p(L, n) = 12,710
With L = 123 and n = 12345, p(L, n) = 3,510,491
With L = 123 and n = 678910, p(L, n) = 193,060,223
1.462 s (752 allocations: 32.19 KiB)
```

## Kotlin

Translation of: Java
```import kotlin.math.ln
import kotlin.math.pow

fun main() {
runTest(12, 1)
runTest(12, 2)
runTest(123, 45)
runTest(123, 12345)
runTest(123, 678910)
}

private fun runTest(l: Int, n: Int) {
//    System.out.printf("p(%d, %d) = %,d%n", l, n, p(l, n))
println("p(\$l, \$n) = %,d".format(p(l, n)))
}

fun p(l: Int, n: Int): Int {
var m = n
var test = 0
val log = ln(2.0) / ln(10.0)
var factor = 1
var loop = l
while (loop > 10) {
factor *= 10
loop /= 10
}
while (m > 0) {
test++
val value = (factor * 10.0.pow(test * log % 1)).toInt()
if (value == l) {
m--
}
}
return test
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12,710
p(123, 12345) = 3,510,491
p(123, 678910) = 193,060,223```

## Mathematica/Wolfram Language

Without compilation arbitrary precision will be used, and the algorithm will be slow, compiled is much faster:

```f = Compile[{{ll, _Integer}, {n, _Integer}},
Module[{l, digitcount, log10power, raised, found, firstdigits, pwr = 2},
l = Abs[ll];
digitcount = Floor[Log[10, l]];
log10power = Log[10, pwr];
raised = -1;
found = 0;
While[found < n,
raised++;
firstdigits = Floor[10^(FractionalPart[log10power raised] + digitcount)];
If[firstdigits == l,
found += 1;
]
];
Return[raised]
]
];
f[12, 1]
f[12, 2]
f[123, 45]
f[123, 12345]
f[123, 678910]
```
Output:
```7
80
12710
3510491
193060223```

## Nim

Translation of: Pascal

This is a translation to Nim of the very efficient Pascal alternative algorithm, with minor adjustments.

```import math, strformat

const
Lfloat64 = pow(2.0, 64)
Log10_2_64 = int(Lfloat64 * log10(2.0))

#---------------------------------------------------------------------------------------------------

func ordinal(n: int): string =
case n
of 1: "1st"
of 2: "2nd"
of 3: "3rd"
else: \$n & "th"

#---------------------------------------------------------------------------------------------------

proc findExp(number, countLimit: int) =

var i = number
var digits = 1
while i >= 10:
digits *= 10
i = i div 10

var lmtLower, lmtUpper: uint64
var log10num = log10((number + 1) / digits)
if log10num >= 0.5:
lmtUpper = if (number + 1) / digits < 10: uint(log10Num * (Lfloat64 * 0.5)) * 2 + uint(log10Num * 2)
else: 0
log10Num = log10(number / digits)
lmtLower = uint(log10Num * (Lfloat64 * 0.5)) * 2 + uint(log10Num * 2)
else:
lmtUpper = uint(log10Num * Lfloat64)
lmtLower = uint(log10(number / digits) * Lfloat64)

var count = 0
var frac64 = 0u64
var p = 0
if lmtUpper != 0:
while true:
inc p
inc frac64, Log10_2_64
if frac64 in lmtLower..lmtUpper:
inc count
if count >= countLimit:
break
else:
# Searching for "999...".
while true:
inc p
inc frac64, Log10_2_64
if frac64 >= lmtLower:
inc count
if count >= countLimit:
break

echo fmt"""The {ordinal(count)} occurrence of 2 raised to a power""" &
fmt""" whose product starts with "{number}" is {p}"""

#———————————————————————————————————————————————————————————————————————————————————————————————————

findExp(12, 1)
findExp(12, 2)

findExp(123, 45)
findExp(123, 12345)
findExp(123, 678910)
```
Output:

Processor: Core I5 8250U

Compilation command: `nim c -d:danger --passC:-flto leading12.nim`

Time: about 110 ms

```The 1st occurrence of 2 raised to a power whose product starts with "12" is 7
The 2nd occurrence of 2 raised to a power whose product starts with "12" is 80
The 45th occurrence of 2 raised to a power whose product starts with "123" is 12710
The 12345th occurrence of 2 raised to a power whose product starts with "123" is 3510491
The 678910th occurrence of 2 raised to a power whose product starts with "123" is 193060223```

## Pascal

First convert 2**i -> 10**x => x= ln(2)/ln(10) *i
The integer part of x is the position of the comma.Only the fraction of x leads to the digits.
0<= base ** frac(x) < base thats 1 digit before the comma
Only the first digits are needed.So I think, the accuracy is sufficient, because the results are the same :-)

```program Power2FirstDigits;

uses
sysutils,
strUtils;

const
{\$IFDEF FPC}
{\$MODE DELPHI}

ld10 :double = ln(2)/ln(10);// thats 1/log2(10)
{\$ELSE}
ld10 = 0.30102999566398119521373889472449;

function Numb2USA(const S: string): string;
var
i, NA: Integer;
begin
i := Length(S);
Result := S;
NA := 0;
while (i > 0) do
begin
if ((Length(Result) - i + 1 - NA) mod 3 = 0) and (i <> 1) then
begin
insert(',', Result, i);
inc(NA);
end;
Dec(i);
end;
end;

{\$ENDIF}

function FindExp(CntLmt, Number: NativeUint): NativeUint;
var
i, cnt, DgtShift: NativeUInt;
begin
//calc how many Digits needed
i := Number;
DgtShift := 1;
while i >= 10 do
begin
DgtShift := DgtShift * 10;
i := i div 10;
end;

cnt := 0;
i := 0;
repeat
inc(i);
// x= i*ld10 -> 2^I = 10^x
// 10^frac(x) -> [0..10[ = exp(ln(10)*frac(i*lD10))
if Trunc(DgtShift * exp(ln(10) * frac(i * lD10))) = Number then
begin
inc(cnt);
if cnt >= CntLmt then
BREAK;
end;
until false;
write('The  ', Numb2USA(IntToStr(cnt)), 'th  occurrence of 2 raised to a power');
write(' whose product starts with "', Numb2USA(IntToStr(Number)));
writeln('" is ', Numb2USA(IntToStr(i)));
FindExp := i;
end;

begin
FindExp(1, 12);
FindExp(2, 12);

FindExp(45, 123);
FindExp(12345, 123);
FindExp(678910, 123);
end.
```
Output:
```The  1th  occurrence of 2 raised to a power whose product starts with "12" is 7
The  2th  occurrence of 2 raised to a power whose product starts with "12" is 80
The  45th  occurrence of 2 raised to a power whose product starts with "123" is 12,710
The  12,345th  occurrence of 2 raised to a power whose product starts with "123" is 3,510,491
The  678,910th  occurrence of 2 raised to a power whose product starts with "123" is 193,060,223
//64Bit real    0m43,031s //32Bit real	0m13,363s```

### alternative

Now only using the fractional part for maximum precision in Uint64
ignoring overflow so frac64 is [0..2**64-1] represent [0..1[
changed trunc(DgtShift*exp(ln(10)*frac(i*lD10))) = Number
No trunc Number/Digits <= .. < (Number+1)/Digits => no trunc
Logarithm (Ln(Number/Digits)/ln(10) <= frac(i*lD10) < ln((Number+1)/Digits)/ln(10) => no exp

```program Power2Digits;
uses
sysutils,strUtils;
const
L_float64 = sqr(sqr(65536.0));//2**64
Log10_2_64 = TRUNC(L_float64*ln(2)/ln(10));

function FindExp(CntLmt,Number:NativeUint):NativeUint;
var
Log10Num : extended;
LmtUpper,LmtLower : UInt64;
Frac64 : UInt64;
i,dgts,cnt: NativeUInt;
begin
i := Number;
dgts := 1;
while i >= 10 do
Begin
dgts *= 10;
i := i div 10;
end;
//trunc is Int64 :-( so '316' was a limit
Log10Num :=ln((Number+1)/dgts)/ln(10);
IF Log10Num >= 0.5 then
Begin
IF (Number+1)/dgts < 10 then
Begin
LmtUpper := Trunc(Log10Num*(L_float64*0.5))*2;
LmtUpper += Trunc(Log10Num*2);
end
else
LmtUpper := 0;
Log10Num :=ln(Number/dgts)/ln(10);
LmtLower := Trunc(Log10Num*(L_float64*0.5))*2;
LmtLower += Trunc(Log10Num*2);
end
Else
Begin
LmtUpper := Trunc(Log10Num*L_float64);
LmtLower := Trunc(ln(Number/dgts)/ln(10)*L_float64);
end;

cnt := 0;
i := 0;
Frac64 := 0;
IF LmtUpper <> 0 then
Begin
repeat
inc(i);
inc(Frac64,Log10_2_64);
IF (Frac64>= LmtLower) AND (Frac64< LmtUpper) then
Begin
inc(cnt);
IF cnt>= CntLmt then
BREAK;
end;
until false
end
Else
//searching for '999..'
Begin
repeat
inc(i);
inc(Frac64,Log10_2_64);
IF (Frac64>= LmtLower) then
Begin
inc(cnt);
IF cnt>= CntLmt then
BREAK;
end;
until false
end;
write('The ',Numb2USA(IntToStr(cnt)),'th  occurrence of 2 raised to a power');
write(' whose product starts with "',Numb2USA(IntToStr(number)));
writeln('" is ',Numb2USA(IntToStr(i)));
FindExp := i;
end;

Begin
FindExp(1,12);
FindExp(2,12);

FindExp(45,223);
FindExp(12345,123);
FindExp(678910,123);

FindExp(1,99);
end.
```
Output:
```The 1th  occurrence of 2 raised to a power whose product starts with "12" is 7
The 2th  occurrence of 2 raised to a power whose product starts with "12" is 80
The 45th  occurrence of 2 raised to a power whose product starts with "223" is 22,670
The 12,345th  occurrence of 2 raised to a power whose product starts with "123" is 3,510,491
The 678,910th  occurrence of 2 raised to a power whose product starts with "123" is 193,060,223
The 1th  occurrence of 2 raised to a power whose product starts with "99" is 93

//64Bit
real	0m0,138s
//32Bit
real    0m0,389s```

## Perl

Translation of: Raku
```use strict;
use warnings;
use feature 'say';
use feature 'state';

use POSIX qw(fmod);
use Perl6::GatherTake;

use constant ln2ln10 => log(2) / log(10);

sub comma { reverse ((reverse shift) =~ s/(.{3})/\$1,/gr) =~ s/^,//r }

sub ordinal_digit {
my(\$d) = \$_ =~ /(.)\$/;
\$d eq '1' ? 'st' : \$d eq '2' ? 'nd' : \$d eq '3' ? 'rd' : 'th'
}

sub startswith12 {
my(\$nth) = @_;
state \$i = 0;
state \$n = 0;
while (1) {
next unless '1.2' eq substr(( 10 ** fmod(++\$i * ln2ln10, 1) ), 0, 3);
return \$i if ++\$n eq \$nth;
}
}

sub startswith123 {
my \$pre = '1.23';
my (\$this, \$count) = (0, 0);

gather {
while (1) {
if (\$this == 196) {
\$this = 289;
\$this = 485 unless \$pre eq substr(( 10 ** fmod((\$count+\$this) * ln2ln10, 1) ), 0, 4);
} elsif (\$this == 485) {
\$this = 196;
\$this = 485 unless \$pre eq substr(( 10 ** fmod((\$count+\$this) * ln2ln10, 1) ), 0, 4);
} elsif (\$this == 289) {
\$this = 196
} elsif (\$this ==  90) {
\$this = 289
} elsif (\$this ==   0) {
\$this = 90;
}
take \$count += \$this;
}
}
}

my \$start_123 = startswith123(); # lazy list

sub p {
my(\$prefix,\$nth) = @_;
\$prefix eq '12' ? startswith12(\$nth) : \$start_123->[\$nth-1];
}

for ([12, 1], [12, 2], [123, 45], [123, 12345], [123, 678910]) {
my(\$prefix,\$nth) = @\$_;
printf "%-15s %9s power of two (2^n) that starts with %5s is at n = %s\n", "p(\$prefix, \$nth):",
comma(\$nth) . ordinal_digit(\$nth), "'\$prefix'", comma p(\$prefix, \$nth);
}
```
Output:
```p(12, 1):             1st power of two (2^n) that starts with  '12' is at n = 7
p(12, 2):             2nd power of two (2^n) that starts with  '12' is at n = 80
p(123, 45):          45th power of two (2^n) that starts with '123' is at n = 12,710
p(123, 12345):   12,345th power of two (2^n) that starts with '123' is at n = 3,510,491
p(123, 678910): 678,910th power of two (2^n) that starts with '123' is at n = 193,060,223```

## Phix

Library: Phix/mpfr
```with javascript_semantics
function p(integer L, n)
atom logof2 = log10(2)
integer places = trunc(log10(L)),
nfound = 0, i = 1
while true do
atom a = i * logof2,
b = trunc(power(10,a-trunc(a)+places))
if L == b then
nfound += 1
if nfound == n then exit end if
end if
i += 1
end while
return i
end function

constant tests = {{12, 1}, {12, 2}, {123, 45}, {123, 12345}, {123, 678910}}
include ordinal.e
include mpfr.e
mpz z = mpz_init()
atom t0 = time()
for i=1 to length(tests)-(2*(platform()=JS)) do
integer {L,n} = tests[i], pln = p(L,n)
mpz_ui_pow_ui(z,2,pln)
integer digits = mpz_sizeinbase(z,10)
string st = iff(digits>2e6?sprintf("%,d digits",digits):
shorten(mpz_get_str(z),"digits",5))
printf(1,"The %d%s power of 2 that starts with %d is %d [i.e. %s]\n",{n,ord(n),L,pln,st})
end for
?elapsed(time()-t0)
```
Output:
```The 1st power of 2 that starts with 12 is 7 [i.e. 128]
The 2nd power of 2 that starts with 12 is 80 [i.e. 1208925819614629174706176]
The 45th power of 2 that starts with 123 is 12710 [i.e. 12338...09024 (3,827 digits)]
The 12345th power of 2 that starts with 123 is 3510491 [i.e. 12317...80448 (1,056,764 digits)]
The 678910th power of 2 that starts with 123 is 193060223 [i.e. 58,116,919 digits]
```

(The last two tests are simply too much for pwa/p2js)

## Python

Using logs, as seen first in the Pascal example.

```from math import log, modf, floor

def p(l, n, pwr=2):
l = int(abs(l))
digitcount = floor(log(l, 10))
log10pwr = log(pwr, 10)
raised, found = -1, 0
while found < n:
raised += 1
firstdigits = floor(10**(modf(log10pwr * raised) + digitcount))
if firstdigits == l:
found += 1
return raised

if __name__ == '__main__':
for l, n in [(12, 1), (12, 2), (123, 45), (123, 12345), (123, 678910)]:
print(f"p({l}, {n}) =", p(l, n))
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12710
p(123, 12345) = 3510491
p(123, 678910) = 193060223```

## Racket

Translation of: Pascal

First implementation is an untyped, naive revision of the Pascal algorithm. But there is a lot of casting between the exact integers an floats. As well as runtime type checking.

Then there is the typed racket version, which is stricter about the use of types (although, I'm not super happy at the algorithm counting using floating-point integers... but it's faster)

Compare and contrast...

### Untyped Racket

```#lang racket
(define (fract-part f)
(- f (truncate f)))

(define ln10 (log 10))
(define ln2/ln10 (/ (log 2) ln10))

(define (inexact-p-test L)
(let ((digit-shift (let loop ((L (quotient L 10)) (shift 1))
(if (zero? L) shift (loop (quotient L 10) (* 10 shift))))))
(λ (p) (= L (truncate (* digit-shift (exp (* ln10 (fract-part (* p ln2/ln10))))))))))

(define (p L n)
(let ((test? (inexact-p-test L)))
(let loop ((j 1) (n (sub1 n)))
(cond [(not (test? j)) (loop (add1 j) n)]
[(zero? n) j]
[else (loop (add1 j) (sub1 n))]))))

(module+ main
(define (report-p L n)
(time (printf "p(~a, ~a) = ~a~%" L n (p L n))))

(report-p 12 1)
(report-p 12 2)
(report-p 123 45)
(report-p 123 12345)
(report-p 123 678910))
```
Output:
```p(12, 1) = 7
cpu time: 1 real time: 1 gc time: 0
p(12, 2) = 80
cpu time: 0 real time: 0 gc time: 0
p(123, 45) = 12710
cpu time: 5 real time: 5 gc time: 0
p(123, 12345) = 3510491
cpu time: 439 real time: 442 gc time: 23
p(123, 678910) = 193060223
cpu time: 27268 real time: 27354 gc time: 194```

### Typed Racket

```#lang typed/racket

(: fract-part (-> Float Float))
(: ln10 Positive-Float)
(: ln2/ln10 Positive-Float)
(: p (-> Positive-Index Positive-Index Positive-Integer))

(define (fract-part f)
(- f (truncate f)))

(define ln10 (cast (log 10) Positive-Float))
(define ln2/ln10 (cast (/ (log 2) ln10) Positive-Float))

(define (inexact-p-test [L : Positive-Index])
(let ((digit-shift : Nonnegative-Float
(let loop ((L (quotient L 10)) (shift : Nonnegative-Float 1.))
(if (zero? L) shift (loop (quotient L 10) (* 10. shift)))))
(l (exact->inexact L)))
(: f (-> Nonnegative-Float Boolean))
(define (f p) (= l (truncate (* digit-shift (exp (* ln10 (fract-part (* p ln2/ln10))))))))
f))

(define (p L n)
(let ((test? (inexact-p-test L)))
(let loop : Positive-Integer ((j : Positive-Float 1.) (n : Index (sub1 n)))
(cond [(not (test? j)) (loop (add1 j) n)]
[(zero? n) (assert (exact-round j) positive?)]
[else (loop (add1 j) (sub1 n))]))))

(module+ main
(: report-p (-> Positive-Index Positive-Index Void))
(define (report-p L n)
(time (printf "p(~a, ~a) = ~a~%" L n (p L n))))

(report-p 12 1)
(report-p 12 2)
(report-p 123 45)
(report-p 123 12345)
(report-p 123 678910))
```
Output:
```p(12, 1) = 7
cpu time: 1 real time: 1 gc time: 0
p(12, 2) = 80
cpu time: 0 real time: 0 gc time: 0
p(123, 45) = 12710
cpu time: 5 real time: 5 gc time: 0
p(123, 12345) = 3510491
cpu time: 237 real time: 231 gc time: 31
p(123, 678910) = 193060223
cpu time: 10761 real time: 10794 gc time: 17```

## Raku

(formerly Perl 6)

Works with: Rakudo version 2019.11

Uses logs similar to Go and Pascal entries. Takes advantage of patterns in the powers to cut out a bunch of calculations.

```use Lingua::EN::Numbers;

constant \$ln2ln10 = log(2) / log(10);

my @startswith12  = ^∞ .grep: { ( 10 ** (\$_ * \$ln2ln10 % 1) ).substr(0,3) eq '1.2' };

my @startswith123 = lazy gather loop {
state \$pre   = '1.23';
state \$count = 0;
state \$this  = 0;
given \$this {
when 196 {
\$this = 289;
my \n = \$count + \$this;
\$this = 485 unless ( 10 ** (n * \$ln2ln10 % 1) ).substr(0,4) eq \$pre;
}
when 485 {
\$this = 196;
my \n = \$count + \$this;
\$this = 485 unless ( 10 ** (n * \$ln2ln10 % 1) ).substr(0,4) eq \$pre;
}
when 289 { \$this = 196 }
when 90  { \$this = 289 }
when 0   { \$this = 90  }
}
take \$count += \$this;
}

multi p (\$prefix where *.chars == 2, \$nth) {  @startswith12[\$nth-1] }
multi p (\$prefix where *.chars == 3, \$nth) { @startswith123[\$nth-1] }

for < 12 1  12 2  123 45  123 12345  123 678910 > -> \$prefix, \$nth {
printf "%-15s %9s power of two (2^n) that starts with %5s is at n = %s\n", "p(\$prefix, \$nth):",
comma(\$nth) ~ ordinal-digit(\$nth).substr(*-2), "'\$prefix'", comma p(\$prefix, \$nth);
}
```
Output:
```p(12, 1):             1st power of two (2^n) that starts with  '12' is at n = 7
p(12, 2):             2nd power of two (2^n) that starts with  '12' is at n = 80
p(123, 45):          45th power of two (2^n) that starts with '123' is at n = 12,710
p(123, 12345):   12,345th power of two (2^n) that starts with '123' is at n = 3,510,491
p(123, 678910): 678,910th power of two (2^n) that starts with '123' is at n = 193,060,223```

## REXX

```/*REXX program computes powers of two whose leading decimal digits are "12" (in base 10)*/
parse arg L n b .                                /*obtain optional arguments from the CL*/
if L=='' | L=="," then L= 12                     /*Not specified?  Then use the default.*/
if n=='' | n=="," then n=  1                     /* "      "         "   "   "     "    */
if b=='' | b=="," then b=  2                     /* "      "         "   "   "     "    */
LL= length(L)                                    /*obtain the length of  L  for compares*/
fd=   left(L, 1)                                 /*obtain the first   dec. digit  of  L.*/
fr= substr(L, 2)                                 /*   "    "  rest of dec. digits  "  " */
numeric digits max(20, LL+2)                     /*use an appropriate value of dec. digs*/
rest= LL - 1                                     /*the length of the rest of the digits.*/
#= 0                                             /*the number of occurrences of a result*/
x= 1                                             /*start with a product of unity (B**0).*/
do j=1  until #==n;        x= x * b         /*raise  B  to a whole bunch of powers.*/
parse var x _ 2                             /*obtain the first decimal digit of  X.*/
if _ \== fd  then iterate                   /*check only the 1st digit at this time*/
if LL>1  then do                            /*check the rest of the digits, maybe. */
\$= format(x, , , , 0)         /*express  X  in exponential format.   */
parse var \$ '.' +1 f +(rest)  /*obtain the rest of the digits.       */
if f \== fr  then iterate     /*verify that  X  has the rest of digs.*/
end                           /* [↓] found an occurrence of an answer*/
#= # + 1                                    /*bump the number of occurrences so far*/
end   /*j*/

say 'The '  th(n)  ' occurrence of '   b  ' raised to a power whose product starts with' ,
' "'L"'"       ' is '        commas(j).
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: arg _;     do c=length(_)-3  to 1  by -3;  _= insert(',', _, c);  end;    return _
th:     arg _;  return _ || word('th st nd rd', 1 +_//10 * (_//100 % 10\==1) * (_//10 <4))
```
output   when using the inputs of:     12   1
```The  1st  occurrence of  2  raised to a power whose product starts with  "12'  is  7.
```
output   when using the inputs of:     12   2
```The  2nd  occurrence of  2  raised to a power whose product starts with  "12'  is  80.
```
output   when using the inputs of:     123   45
```The  45th  occurrence of  2  raised to a power whose product starts with  "123'  is  12,710.
```
output   when using the inputs of:     123   12345
```The  12345th  occurrence of  2  raised to a power whose product starts with  "123'  is  3,510,491.
```
output   when using the inputs of:     123   678910
```The  678910th  occurrence of  2  raised to a power whose product starts with  "123'  is  193,060,223.
```

## Ruby

Translation of: C
```def p(l, n)
test = 0
logv = Math.log(2.0) / Math.log(10.0)
factor = 1
loopv = l
while loopv > 10 do
factor = factor * 10
loopv = loopv / 10
end
while n > 0 do
test = test + 1
val = (factor * (10.0 ** ((test * logv).modulo(1.0)))).floor
if val == l then
n = n - 1
end
end
return test
end

def runTest(l, n)
print "P(%d, %d) = %d\n" % [l, n, p(l, n)]
end

runTest(12, 1)
runTest(12, 2)
runTest(123, 45)
runTest(123, 12345)
runTest(123, 678910)
```
Output:
```P(12, 1) = 7
P(12, 2) = 80
P(123, 45) = 12710
P(123, 12345) = 3510491
P(123, 678910) = 193060223```

## Rust

```fn power_of_two(l: isize, n: isize) -> isize {
let mut test: isize = 0;
let log: f64 = 2.0_f64.ln() / 10.0_f64.ln();
let mut factor: isize = 1;
let mut looop = l;
let mut nn = n;
while looop > 10 {
factor *= 10;
looop /= 10;
}

while nn > 0 {
test = test + 1;
let val: isize = (factor as f64 * 10.0_f64.powf(test as f64 * log % 1.0)) as isize;

if val == l {
nn = nn - 1;
}
}

test
}

fn run_test(l: isize, n: isize) {
println!("p({}, {}) = {}", l, n, power_of_two(l, n));
}

fn main() {
run_test(12, 1);
run_test(12, 2);
run_test(123, 45);
run_test(123, 12345);
run_test(123, 678910);
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12710
p(123, 12345) = 3510491
p(123, 678910) = 193060223
```

## Scala

Translation of: Java
```object FirstPowerOfTwo {
def p(l: Int, n: Int): Int = {
var n2 = n
var test = 0
val log = math.log(2) / math.log(10)
var factor = 1
var loop = l
while (loop > 10) {
factor *= 10
loop /= 10
}
while (n2 > 0) {
test += 1
val value = (factor * math.pow(10, test * log % 1)).asInstanceOf[Int]
if (value == l) {
n2 -= 1
}
}
test
}

def runTest(l: Int, n: Int): Unit = {
printf("p(%d, %d) = %,d%n", l, n, p(l, n))
}

def main(args: Array[String]): Unit = {
runTest(12, 1)
runTest(12, 2)
runTest(123, 45)
runTest(123, 12345)
runTest(123, 678910)
}
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12,710
p(123, 12345) = 3,510,491
p(123, 678910) = 193,060,223```

## Sidef

```func farey_approximations(r, callback) {

var (a1 = r.int, b1 = 1)
var (a2 = a1+1,  b2 = 1)

loop {
var a3 = a1+a2
var b3 = b1+b2

if (a3 < r*b3) {
(a1, b1) = (a3, b3)
}
else {
(a2, b2) = (a3, b3)
}

callback(a3 / b3)
}
}

func p(L, nth) {

define ln2  = log(2)
define ln5  = log(5)
define ln10 = log(10)

var t = L.len-1

func isok(n) {
floor(exp(ln2*(n - floor((n*ln2)/ln10) + t) + ln5*(t - floor((n*ln2)/ln10)))) == L
}

var deltas = gather {
farey_approximations(ln2/ln10, {|r|
take(r.de) if (r.de.len == L.len)
break      if (r.de.len >  L.len)
})
}.sort.uniq

var c = 0
var k = (1..Inf -> first(isok))

loop {
return k if (++c == nth)
k += (deltas.first {|d| isok(k+d) } \\ die "error: #{k}")
}
}

var tests = [
[12, 1],
[12, 2],
[123, 45],
[123, 12345],
[123, 678910],

# extra
[1234, 10000],
[12345, 10000],
]

for a,b in (tests) {
say "p(#{a}, #{b}) = #{p(a,b)}"
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12710
p(123, 12345) = 3510491
p(123, 678910) = 193060223
p(1234, 10000) = 28417587
p(12345, 10000) = 284166722
```

## Swift

Translation of: Go

### Slower Version

```let ld10 = log(2.0) / log(10.0)

func p(L: Int, n: Int) -> Int {
var l = L
var digits = 1

while l >= 10 {
digits *= 10
l /= 10
}

var count = 0
var i = 0

while count < n {
let rhs = (Double(i) * ld10).truncatingRemainder(dividingBy: 1)
let e = exp(log(10.0) * rhs)

if Int(e * Double(digits)) == L {
count += 1
}

i += 1
}

return i - 1
}

let cases = [
(12, 1),
(12, 2),
(123, 45),
(123, 12345),
(123, 678910)
]

for (l, n) in cases {
print("p(\(l), \(n)) = \(p(L: l, n: n))")
}
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12710
p(123, 12345) = 3510491
p(123, 678910) = 193060223```

### Faster Version

```import Foundation

func p2(L: Int, n: Int) -> Int {
let asString = String(L)
var digits = 1

for _ in 1...18-asString.count {
digits *= 10
}

let ten18 = Int(1e18)

var count = 0, i = 0, probe = 1

while true {
probe += probe
i += 1

if probe >= ten18 {
while true {
if probe >= ten18 {
probe /= 10
}

if probe / digits == L {
count += 1

if count >= n {
count -= 1
break
}
}

probe += probe
i += 1
}
}

let probeString = String(probe)
var len = asString.count

if asString.count > probeString.count {
len = probeString.count
}

if probeString.prefix(len) == asString {
count += 1

if count >= n {
break
}
}
}

return i
}

let cases = [
(12, 1),
(12, 2),
(123, 45),
(123, 12345),
(123, 678910)
]

for (l, n) in cases {
print("p(\(l), \(n)) = \(p2(L: l, n: n))")
}
```
Output:

Same as before.

## Visual Basic .NET

```Module Module1

Function Func(ByVal l As Integer, ByVal n As Integer) As Long
Dim res As Long = 0, f As Long = 1
Dim lf As Double = Math.Log10(2)
Dim i As Integer = l

While i > 10
f *= 10
i /= 10
End While

While n > 0
res += 1

If CInt((f * Math.Pow(10, res * lf Mod 1))) = l Then
n -= 1
End If
End While

Return res
End Function

Sub Main()
Dim values = {Tuple.Create(12, 1), Tuple.Create(12, 2), Tuple.Create(123, 45), Tuple.Create(123, 12345), Tuple.Create(123, 678910), Tuple.Create(99, 1)}
For Each pair In values
Console.WriteLine("p({0,3}, {1,6}) = {2,11:n0}", pair.Item1, pair.Item2, Func(pair.Item1, pair.Item2))
Next
End Sub

End Module
```
Output:
```p( 12,      1) =          60
p( 12,      2) =          70
p(123,     45) =      12,710
p(123,  12345) =   3,496,509
p(123, 678910) = 192,278,374
p( 99,      1) =          93```

## Wren

Translation of: Go
Library: Wren-fmt
Library: Wren-math

Just the first version which has turned out to be much quicker than the Go entry (around 28 seconds on the same machine). Frankly, I've no idea why.

```import "/fmt" for Fmt
import "/math" for Math

var ld10 = Math.ln2 / Math.ln10

var p = Fn.new { |L, n|
var i = L
var digits = 1
while (i >= 10) {
digits = digits * 10
i = (i/10).floor
}
var count = 0
i = 0
while (count < n) {
var e = (Math.ln10 * (i * ld10).fraction).exp
if ((e * digits).truncate == L) count = count + 1
i = i + 1
}
return i - 1
}

var start = System.clock
var params = [ [12, 1] , [12, 2], [123, 45], [123, 12345], [123, 678910] ]
for (param in params) {
Fmt.print("p(\$d, \$d) = \$,d", param, param, p.call(param, param))
}

System.print("\nTook %(System.clock - start) seconds.")
```
Output:
```p(12, 1) = 7
p(12, 2) = 80
p(123, 45) = 12,710
p(123, 12345) = 3,510,491
p(123, 678910) = 193,060,223

Took 16.51308 seconds.
```

## Yabasic

```FAC = 0.30102999566398119521373889472449302677

print p(12, 1)
print p(12, 2)
print p(123, 45)
print p(123, 12345)
print p(123, 678910)
end

sub p(L, n)
cont = 0 : j = 0
LS\$ = str\$(L)
while cont < n
j = j + 1
x = FAC * j
//if x < len(LS\$)  continue while       'sino da error
y = 10^(x-int(x))
y = y * 10^len(LS\$)
digits\$ = str\$(y)
if left\$(digits\$, len(LS\$)) = LS\$  cont = cont + 1
end while
return j
end sub```

## zkl

Translation of: Pascal

Lots of float are slow so I've restricted the tests.

```// float*int --> float and int*float --> int
fcn p(L,nth){   // 2^j = <L><digits>
var [const] ln10=(10.0).log(), ld10=(2.0).log() / ln10;
digits := (10).pow(L.numDigits - 1);
foreach i in ([1..]){
z:=ld10*i;
if(L == ( ln10 * (z - z.toInt()) ).exp()*digits and (nth-=1) <= 0)
return(i);
}
}```
Library: GMP
GNU Multiple Precision Arithmetic Library

GMP is just used to give some context on the size of the numbers we are dealing with.

```var [const] BI=Import("zklBigNum");  // libGMP
tests:=T( T(12,1),T(12,2), T(123,45),T(123,12345), );
foreach L,nth in (tests){
n:=p(L,nth);
println("2^%-10,d is occurance %,d of 2^n == '%d<abc>' (%,d digits)"
.fmt(n,nth,L,BI(2).pow(n).len()));
}```
Output:
```2^7          is occurance 1 of 2^n == '12<abc>' (3 digits)
2^80         is occurance 2 of 2^n == '12<abc>' (25 digits)
2^12,710     is occurance 45 of 2^n == '123<abc>' (3,827 digits)
2^3,510,491  is occurance 12,345 of 2^n == '123<abc>' (1,056,764 digits)
```