Task: First power of 2 that has leading decimal digits of 12
C#: 

using System;

class Program {

    
    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;
    }

    
    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;
    }

    
    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);
        
        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 
            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);
    }
}

Go: 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 := [][2]uint64{{12, 1}, {12, 2}, {123, 45}, {123, 12345}, {123, 678910}}
    for _, param := range params {
        fmt.Printf("p(%d, %d) = %s\n", param[0], param[1], commatize(p(param[0], param[1])))
    }
    fmt.Printf("\nTook %s\n", time.Since(start))
}