Logarithm Questions with Solutions

If ${log}_{3}2=x$ and ${log}_{3}5=y$, find ${log}_{3}1620$.

If ${log}_{5}2=x$, then find ${log}_{25}800$.

The logarithm of the difference between the squares of two positive integers $A$ and $B$ is a fixed constant $C$. If $A=B$, then the value of $C$ is

Which of the following numbers is the coefficient of $x^{100}$ in the expansion of ${log}_e\left(\frac{1+x}{1+x^2}\right)$, $\left|x\right|<1$?

The solution set of ${log}_3\left(\left(x+2\right)\left(x+4\right)\right)+{log}_{1/3}\left(x+2\right)<\frac{1}{2}{log}_{\sqrt{3}}7$ is

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

The value of $y=0.36{log}_{0.25}\left(\frac{1}{3}+\frac{1}{3^2}+\cdots to\infty \right)$ is

For $a>0,a\ne 1$, the number of values of $x$ satisfying $2{log}_{x}a+{log}_{ax}a+3{log}_{a^{2}x}a=0$ is

If $log\left(1-x+x^2\right)=a_{1}x+a_2{x}^2+a_3{x}^3+\dots$, then $a_3+a_6+a_9+\dots$ is equal to

The value of $3^{3-{log}_{3}5}$ is

If $\left(x_0,y_0\right)$ is the solution of the equations ${\left(2x\right)}^{ln2}={\left(3y\right)}^{ln3}$ and $3^{lnx}=2^{lny}$, then $x_0$ is:

If $x$, $y$, $z$ are three consecutive positive integers, then $log\left(1+xz\right)$ is

If ${log}_{3}2$, ${log}_3\left(2^x-5\right)$, ${log}_3\left(2^x-\frac{7}{2}\right)$ are in arithmetic progression, then the value of x is equal to

If $8^{x-1}={\left(\frac{1}{4}\right)}^x$, then the value of $\frac{1}{{log}_{x+1}4-{log}_{x+1}5}+\frac{1}{{log}_{1-x}4-{log}_{1-x}5}$ is

Let $x$ be a positive real number such that $x^{8{log}_{5}x-24}=5^{-4}$. Then the product of all possible values of $x$ is