SOLUTION: Prove that {{{ a^b = b^a }}} with a= 2 1/4 and b= 3 3/8.

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Question 1065452: Prove that +a%5Eb+=+b%5Ea+ with a= 2 1/4 and b= 3 3/8.
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Sub 2.25 for a and 3.375 for b.
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See if it's true.

Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.
Prove that +a%5Eb+=+b%5Ea+ with a= 2 1/4 and b= 3 3/8.
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Notice that a = 9%2F4 and b = 27%2F8.

Take logarithms of both sides. 

log%28%28a%5Eb%29%29 = b%2Alog%28%28a%29%29 = %2827%2F8%29%2Alog%28%289%2F4%29%29 = 27%2F8%29%2Alog%28%28%283%2F2%29%29%5E2%29 = %2827%2F8%29%2A2%2Alog%28%283%2F2%29%29 = %2827%2F4%29%2Alog%28%283%2F2%29%29.


log%28%28b%5Ea%29%29 = a%2Alog%28%28b%29%29 = %289%2F4%29%2Alog%28%2827%2F8%29%29 = 9%2F4%29%2Alog%28%28%283%2F2%29%29%5E3%29 = %289%2F4%29%2A3%2Alog%28%283%2F2%29%29 = %2827%2F4%29%2Alog%28%283%2F2%29%29.


We just got that the logarithms are the same.


Hence, the under-the-logarithm expressions are equal.

Proved and solved.