Single Variable Calculus, Chapter 7, 7.4-2, Section 7.4-2, Problem 8

Find the expression a.) $\log_{10} \sqrt{10}$ and b.) $\log_8 320 - \log_8 5$

a.) $\log_{10} \sqrt{10}$


$
\begin{equation}
\begin{aligned}

&
&& \text{Log Laws}
\\
\\
\log_{10} \sqrt{10} =& \frac{1}{2} \log_{10} 10
&& \log_a^b = b \log a
\\
\\
\log_{10} \sqrt{10} =& \frac{1}{2} (1)
&& \log_a a
\\
\\
\log_{10} \sqrt{10} =& \frac{1}{2}
&&

\end{aligned}
\end{equation}
$




b.) $\log_8 320 - \log_8 5$


$
\begin{equation}
\begin{aligned}

&
&& \text{Log Laws}
\\
\\
\log_8 320 - \log_8 5 =& \log_8 \left( \frac{320}{5} \right)
&& \log \left( \frac{a}{b} \right) = \log a - \log b
\\
\\
\log_8 320 - \log_8 5 =& \log_8 64
&&
\\
\\
\log_8 320 - \log_8 5 =& \log_8 (8)^2
&& \log a^b - b \log a
\\
\\
\log_8 320 - \log_8 5 =& 2 \log_8 8
&& \log_a a = 1
\\
\\
\log_8 320 - \log_8 5 =& 2(1)
&&
\\
\\
\log_8 320 - \log_8 5 =& 2
&&

\end{aligned}
\end{equation}
$