SOLUTION: log_3(x^(log

SOLUTION: log_3(x^(log_3(x))=4

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.

RELATED QUESTIONS
log(4+x)-log(x-3)=log2 (answered by farah)

log(x-4) = -2 +... (answered by jsmallt9,LinnW)

log(x^3)+log(x^4)/log(80x)=3 (answered by ewatrrr)

Log(3) 5 + Log(3) 4 = Log(x)... (answered by Alan3354)

solve exactly log(x+4)-log(x-3)=log... (answered by jim_thompson5910,MathLover1)

log 3 x= log 5 + log... (answered by solver91311)

log(x + 4) + log(x + 3) = log... (answered by dabanfield)

log (2+x)-log (x-4)=log 3 (answered by nerdybill)

log(x+3)+log(x+4)=log(X3+13x2)-logx (answered by ikleyn)

log(x + 3) = log 4 − log(2x −... (answered by MathLover1)