SOLUTION: solve for n: [(3)]^{(81)^n}=[(81)]^{(3)^n}

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Question 935849: solve for n:
[(3)]^{(81)^n}=[(81)]^{(3)^n}
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
highlight%283%5E%2881%5En%29=81%5E%283%5En%29%29 ...........81=9%2A9=3%5E2%2A3%5E2=3%5E4 substitute it

highlight%283%5E%28%283%5E4%29%5En%29=%283%5E4%29%5E%283%5En%29%29

highlight%283%5E%283%5E%284n%29%29=3%5E%284%2A3%5En%29%29 ....same base, then same exponents

highlight%283%5E%284n%29=4%283%29%5En%29

highlight%283%5E%284n%29%2F3%5En=4%29

highlight%283%5E%284n-n%29=4%29

highlight%283%5E%283n%29=4%29 ....use log base 10 to solve for n

highlight%28log%283%5E%283n%29%29=log%284%29%29

highlight%28%283n%29log%283%29=log%282%5E2%29%29

highlight%283n=2log%282%29%2Flog%283%29%29

highlight%28blue%28n+=+%282log%282%29%29%2F%283log%283%29%29%29%29...exact solution

highlight%28green%28n=0.42062%29%29 ...approximate solution

check this solution:

highlight%283%5E%2881%5E%280.42062%29%29=+81%5E%283%5E%280.42062%29%29%29+

highlight%281070.37=1070.36%29+

highlight%281070=1070%29+