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.
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