Solve the exponential equations:

\(10^{2x}=50\)

\(8\cdot 3^{x}=5\)

that logarithms to the base \(10\) are called common logarithms and are written without a subscript. That is, \(x=\log _{10}{y}\) is written \(x=\log y\).

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. \[ x=\log _{3} \dfrac{5}{8} =\dfrac{\ln \dfrac{5}{8} }{\ln 3}\approx -0.428 \]