Solving an equaiton which includes $log$ as both base and exponent

Question

Q: If $$9x = x^{\log_3x}$$ then what is $x$ ?

I can't solve it. I have tried to use identities in my book but i think they are useless for this question. I need a hint

Try taking $\log_3$ of both sides – Henry Apr 23 '15 at 13:38 — Henry, Apr 23 '15 at 13:38

score 2 · Answer 1 · answered Apr 23 '15 at 13:37

2

Hint: Try taking logarithm to the base $3$ on both sides. You'll get a quadratic in $\log_3x$.

answered Apr 23 '15 at 13:37

Arpan

2,633

score 0 · Accepted Answer · answered Apr 23 '15 at 13:38

0

It is $$\log_3(9x)=(\log_3x)^2$$

Can you proceed from here?

answered Apr 23 '15 at 13:38

Paul

20,553

Yes I can. Thank you very much – Oğuz İsmayil uysal Apr 23 '15 at 13:40
@Oğuzİsmayiluysal: You are welcome. – Paul Apr 23 '15 at 13:41

score -1 · Answer 3 · answered Apr 23 '15 at 13:39

Take $\log_3$ of both sides. Then $$ \log_3{9} + \log_3{x} = (\log_3{x}) (\log_3{x}), $$ using $\log{ab}=\log{a}+\log{b}$ and $\log{a^b} = b\log{a}$. This is a quadratic equation for $\log_3{x}$. Solve that and exponentiate $3$ with the answer to find $x$.

Solving an equaiton which includes $log$ as both base and exponent

3 Answers3