If $p^3 = q^4 = r^6 = t^7 = s^2$,then $\log_t(pqrs) = \ldots$.

The expression $\log_{a} x$ is defined for $(a > 0, a \neq 1)$ when:

Let $a, b, x$ be positive real numbers with $a \neq 1, x \neq 1, ab \neq 1$. Suppose $\log_{a} b = 10$,and $\frac{\log_{a} x \cdot \log_{x}(\frac{b}{a})}{\log_{x} b \cdot \log_{ab} x} = \frac{p}{q}$,where $p$ and $q$ are positive integers which are coprime. Then $p+q$ is

For what values of $x$ is the following identity valid and holds? $\tanh^{-1}(x) = \frac{1}{2} \log_e \left( \frac{1+x}{1-x} \right)$.

The number of integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^2 \geq 0$ is

Below are four equations in $x$. Assume that $0 < r < 4$. Which of the following equations has the largest solution for $x$?