First power of 2 that has leading decimal digits of 12

(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 n^th-smallest value of j such that the base ten representation of 2^j 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

Task

find:

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

display the results here, on this page.

REXX

<lang 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))</lang>

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.