College Algebra, Chapter 5, 5.4, Section 5.4, Problem 24

Determine the solution of the exponential equation $7^{\frac{x}{2}} = 5^{1-x}$.

$\begin{equation} \begin{aligned} 7^{\frac{x}{2}} &= 5^{1-x}\\ \\ \log 7^{\frac{x}{2}} &=\log 5^{1-x} && \text{Take $\log$ of each side}\\ \\ \frac{x}{2} \log 7 &= (1-x) \log 5 && \text{Law of Logarithms } \log_a A^c = C \log_a A\\ \\ \frac{x}{2} \frac{\log 7}{\log 5} &= 1 - x && \text{Divide by } \log 5\\ \\ \frac{x\log7}{x\log 5} + x &= 1 && \text{Add } x\\ \\ x \left( \frac{\log 7}{2 \log 5} + 1 \right) &= 1 && \text{Factor out } x\\ \\ x &= \frac{1}{\left( \frac{\log 7}{2 \log 5} + 1 \right)} && \text{Divide by } \frac{\log 7}{2 \log 5} + 1\\ \\ x &= 0.6232 \end{aligned} \end{equation}$