Solve the exponential equations:

Solution (a) Since $10$ and $50$ cannot be written with the same base, we write the exponential equation as a logarithm. $$10^{2x}=50\quad \hbox{if and only if}\quad \log 50=2x$$ Then $x=\dfrac{\log 50}{2}$ is an exact solution of the equation. Using a calculator, an approximate solution is $x=\dfrac{\log 50}{2}\approx 0.849$.

(b) It is impossible to write $8$ and $5$ as a power of $3$, so we write the exponential equation as a logarithm. $$\begin{eqnarray*} 8\cdot 3^{x} &=&5 \\[5pt] 3^{x} &=&\dfrac{5}{8} \\[5pt] \log _{3} \dfrac{5}{8} &=&x \end{eqnarray*}$$ Now we use the change-of-base formula to obtain the exact solution of the equation. An approximate solution can then be obtained using a calculator. $$x=\log _{3} \dfrac{5}{8} =\dfrac{\ln \dfrac{5}{8} }{\ln 3}\approx -0.428$$