log

solve


$(\log_3{(4x^2-9)})^{ \log_3{(x+6)} }=9$

guessing.

How?

Don't you mean "inspection"?

No, I mean by examination of integer values, $n \in \mathbb{Z}$, on a closed interval such that the single-valued function exhibits injective behaviour and is trivially evaluated using said integer values.

Ohh I do apologise, I didn't notice the extra set of brackets. Post removed!

[I just had my last university exam this morning, cut me some slack ]

i swear to God these threads are just questions drmath can't do on his problem sheets for uni.

he should at least have some input.

plus, koyla's answer works, what makes you think there are more roots?

It would be good to prove that x = 3 is the only solution. I have tried proving that the function on the left of the equation is increasing, but I got nowhere.

It's certainly increasing when $x > \sqrt{3}$, so we just need to consider what happens over the interval $\frac{3}{2} < x < \sqrt{3}$. As neither $log _3 (4x^2 - 9)$ or $log _3 (x+6)$ take integer values over that interval, we can conclude there are no solutions in that interval and so $x=3$ is the only solution.