SOLUTION: please help me prove that- {{{16^log3 = 9^log4}}}

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Question 481432: please help me prove that-
16%5Elog3+=+9%5Elog4
Answer by kingme18(98) About Me  (Show Source):
You can put this solution on YOUR website!
First, note that both bases are perfect squares: 16=4%5E2 and 9=3%5E2. Let's rewrite your equation:
16%5Elog3+=+9%5Elog4
%284%5E2%29%5Elog3=%283%5E2%29%5Elog4
Recall that when you raise a power to a power, you multiply the exponents:
4%5E%282%2Alog3%29=3%5E%282%2Alog4%29
We are allowed to take the logarithm of both sides of the equation:
log4%5E%282%2Alog3%29=log3%5E%282%2Alog4%29
My favorite property of logarithms is that you can "kick the exponent to the front" :)
2%2Alog3%2Alog4=2%2Alog4%2Alog3
Rearrange to make it obvious (commutative property):
2%2Alog3%2Alog4=2%2Alog3%2Alog4