If $x+\log _{15}\left(5+3^x\right)=x \log _{15} 5+\log _{15} 24$,then $x=\ldots .$.

The equation $x^{(\log _{3} x)^{2}-\frac{9}{2} \log _{3} x+5}=3 \sqrt{3}$ has

Let $n$ be a positive integer such that $\log _2 \log _2 \log _2 \log _2 \log _2(n) < 0 < \log _2 \log _2 \log _2 \log _2(n)$. Let $l$ be the number of digits in the binary expansion of $n$. Then the minimum and the maximum possible values of $l$ are

The value of $(0.16)^{\log _{2.5}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots \infty\right)}$ is equal to

The number of real values of the parameter $k$ for which the equation $(\log_{16} x)^2 - \log_{16} x + \log_{16} k = 0$ has exactly one real solution is:

The number of digits in $20^{301}$ (given,$\log _{10} 2=0.3010$) is: