SOLUTION: If ( 64/729 )^x equals 7/19/32, or 243/32, then what is the value of x?

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Question 1131406: If ( 64/729 )^x equals 7/19/32, or 243/32, then what is the value of x?
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

%28+64%2F729+%29%5Ex+=+7%2F%2819%2F32%29
%28+64%2F729+%29%5Ex+=+224%2F19
log%28%28+64%2F729+%29%5Ex%29+=+log%28%28224%2F19%29%29
x%2Alog%28%28+64%2F729+%29%29+=+log%28%28224%2F19%29%29
x=+log%28%28224%2F19%29%29%2Flog%28%28+64%2F729+%29%29+
x» -1.014147


or 243%2F32, then what is the value of x
%28+64%2F729+%29%5Ex+=+243%2F32
log%28%28+64%2F729+%29%5Ex%29+=+log%28%28243%2F32%29%29
x%2Alog%28%28+64%2F729+%29%29+=+log%28%28243%2F32%29%29
x=+log%28%28243%2F32%29%29%2Flog%28%28+64%2F729+%29%29+
x=+-5%2F6


Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


%2864%2F729%29%5Ex+=+243%2F32

Observe that 64, 729, 243, and 32 are all powers of 2 or 3....

%28%282%2F3%29%5E6%29%5Ex+=+%283%2F2%29%5E5
%282%2F3%29%5E%286x%29+=+%282%2F3%29%5E%28-5%29

The bases are the same, so the exponents are equal:

6x+=+-5

x+=+-5%2F6