The number of solutions of the equation $\log _{(x+1)}(2 x^{2}+7 x+5)+\log _{(2 x+5)}(x+1)^{2}-4=0$ for $x > 0$ is:

If $x, y, z$ are the sides of a right-angled triangle, where $z$ is the hypotenuse (largest side), then find the value of $\frac{1}{\log_{x+z} y} + \frac{1}{\log_{z-x} y}$.

Let $(x_0, y_0)$ be the solution of the following equations: $(2x)^{\ln 2} = (3y)^{\ln 3}$ and $3^{\ln x} = 2^{\ln y}$. Then $x_0$ is

Given that $a, b \in \{0, 1, 2, \ldots, 9\}$ with $a+b \neq 0$ and that $\left(a+\frac{b}{10}\right)^x = \left(\frac{a}{10}+\frac{b}{100}\right)^y = 1000$. Then, $\frac{1}{x} - \frac{1}{y}$ is equal to

If $x = \log_a (bc)$, $y = \log_b (ca)$, and $z = \log_c (ab)$, then the value of $\frac{1}{1+x} + \frac{1}{1+y} + \frac{1}{1+z}$ is

If $\cosh ^{-1}\left(\frac{5}{3}\right)+\sinh ^{-1}\left(\frac{3}{4}\right)=k$, then $e^k=$