Question 886289: log_3(x^(log_3(x))=4
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! best i can do with this is as follows:
log3(x^(log3(x)) = 4
since log(a^b) = b*log(a), this equation becomes equivalent to:
log3(x) * log3(x) = 4
this is equivalent to:
(log3(x))^2 = 4
take the square root of both sides and you get:
log3(x) = 2
this is true if and only if 3^2 = x which makes x = 9
assuming x = 9, your original equation becomes:
log3(9^(log3(9)) = 4
log3(9) = y if and only if 3^y = 9
this makes y = 2
this makes log3(9) equal to 2
equation of log3(9^(log3(9)) = 4 becomes:
log3(9^2) = 4 which becomes:
log3(81) = 4
log3(81) = y if and only if 3^y = 81.
thi makes y = 4 since 3^4 = 81
your equation of log3(81) = 4 becomes 4 = 4.
this confirms the solution of x = 9 is correct.
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