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There is a unique positive real number x such that the three numbers log_8(2x), log_4(x), and log_2(x), in that order, form a geometric progression with a positive common ratio.

The number x can be written as m/n, where m and n are relatively prime positive integers.

Find m+n.

May 27, 2020

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log8(2x),  log4(x),  and  log2(x)  form a geometric progression, find the value of  x.

First, let's write each of these expressions with a base of 2 (using the change-of-base formula):

log8(2x)  =  log2(2x) / log2(8)

=  log2(2x) / 3                                      (because log2(8) = 3)

=  [ log2(2) + log2(x) ] / 3                     (multiplying numbers = adding logs)

=  [ 1 + log2(x) ] / 3                              (because log2(2) = 1)

log4(x)  =  log2(x) / log2(4)

=  log2(x) / 2                                            (because log2(4)  2)

The common ratio can be found by dividing a term by the next term; I'm going to divide the second term

by the third term:  log2(x) / [ log2(x) / 2 ]  =  2   --->   the common ratio is 2.

This means that I can multiply the first term by  2  to get the second term:

2 · [ 1 + log2(x) ] / 3   =  log2(x) / 2

4 · [ 1 + log2(x) ] / 3   =  log2(x)             (multiply both sides by 4)

4/3  + 4/3 · log2(x)  =  log2(x)             (distribute the 4)

4/3  =  -1/3· log2(x)     (subtract  4/3 · log2(x)  from both sides)

-4  =  log2(x)             (multiply both sides by -3)

2-4  =  x                     (change from log form into exponential form)

x  =  1/16

May 27, 2020