Logarithms

For $a>0,\ a\ne1$, the number of values of $x$ satisfying $2\log_x a+\log_{ax} a+3\log_{a^2x} a=0$ is

The value of $y=0.36\log_{0.25}\left(\dfrac13+\dfrac1{3^2}+\cdots\right)$ is

Find the value of $x$, if: $\left( 2^{\frac{1}{\log_x 4}} \right) \left( 2^{\frac{1}{\log_x 16}} \right) \left( 2^{\frac{1}{\log_x 256}} \right) \cdots = 2$

$a,b,c$ are positive and $c>a$ and in H.P. Compute $\log(a+c)+\log(a-2b+c)$.

Solve inequality $\log_3\big((x+2)(x+4)\big)+\log_{1/3}(x+2)<\dfrac12\log_{\sqrt{3}}7$

If $log_x^y=100$ and $log_2^x=10$ then the value of y is.

If a, b, c are in geometric progression, then $log_{ax}^{a}, log_{bx}^{a}$ and $log_{cx}^{a}$ are in