Find solutions of the inequality Find all real values of $x$ satisfying the inequality: $x^{\log_3
 (x+4)} \lt 3^5$.
Hint: take the logarithm w.r.t to $3$ on both sides of the inequality.
We need to solve $f(x)&lt;5$, where $f(x)=\log_3x\log_3(x+4)$.

But $f$ is an increasing function on $[1,+\infty)$ and all $0&lt;x&lt;1$ are solutions.

Thus, the answer is $(0,x_0)$, where $x_0$ is a root of the equation $f(x)=5$:

$x_0=9.9024...$