Question

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
$\ln \frac{e^5}{2}$

Short answer

The solution is$5-\ln \left(2\right)$.

Step-by-step solution

Step 1. Given and explanation.

We have $\ln \frac{e^5}{2}$.

The quotient property states that if there is an expression as ${\log }_a\frac{m}{n}$, it is equal to ${\log }_{a}m-{\log }_{a}n$.

Applying this, we get the solution.

Step 2. Finding the solution.

We have $\ln \frac{e^5}{2}$.

Applying the quotient property, we get
$$\begin{gathered} \ln \frac{e^5}{2}=\ln \left(e^5\right)-\ln \left(2\right) \\ =5-\ln \left(2\right) \end{gathered}$$
$\ln {\left(e\right)}^5=5$ because
$\ln {\left(e\right)}^x=x$. Thus we get the value.