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}
$