Solve the equation: $\;$ $5^{\log x} = 50 - x^{\log 5}$
Given equation: $\;\;$ $5^{\log x} = 50 - x^{\log 5}$ $\;\;\; \cdots \; (1)$
Let $\;$ $5^{\log x} = p$
Then, by definition $\;$ $\log_5 p = \log x$ $\;\;\; \cdots \; (2)$
$\implies$ $x = 10^{\log_5 p}$ $\;\;\; \cdots \; (3)$
In view of equation $(3)$, equation $(1)$ becomes
$p = 50 - \left(10^{\log_5 p}\right)^{\log 5}$
i.e. $\;$ $p = 50 - \left[10^{\left(\frac{\log p}{\log 5}\right)}\right]^{\log 5}$
i.e. $\;$ $p = 50 - 10^{\log p}$
i.e. $\;$ $p = 50 - p$
i.e. $\;$ $2p = 50$ $\implies$ $p = 25$
Substituting the value of $p$ in equation $(2)$ gives
$\log_5 25 = \log x$
i.e. $\;$ $\log x = \log_5 5^2 = 2 \log_5 5 = 2$
$\implies$ $x = 10^2 = 100$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{100 \right\}$