Question

Evaluating Logarithms Evaluate the expression.

(a)${\log }_{8}0.25$ (b) $\ln e^4$ (c)$\ln \left(\frac{1}{e}\right)$

Short answer

1. The required value is $-\frac{2}{3}$.
2. The required value is4.
3. The required value is $-1$.

Step-by-step solution

Part a. Step 1. Given.

The given expression is ${\log }_{8}0.25$.

Part a. Step 2. To determine.

We have to evaluate the given expression.

Part a. Step 3. Calculation.

$$\begin{gathered} {\log }_{8}0.25 \\ ={\log }_8\left(\frac{25}{100}\right) \\ ={\log }_8\left(\frac{1}{4}\right) \\ ={\log }_8\left(2^{-2}\right) \\ ={\log }_{2^3}\left(2^{-2}\right) \\ =\frac{-2}{3}{\log }_2\left(2\right)[\mathrm{b}\text{ecause\hspace{0.17em}\hspace{0.17em}}{\log }_{a^n}\left(b^m\right)=\frac{m}{n}{\log }_a\left(b\right)] \\ =-\frac{2}{3}\left(1\right)[\text{because\hspace{0.17em}\hspace{0.17em}}{\log }_a\left(a\right)=1] \\ =-\frac{2}{3} \end{gathered}$$.

Hence, ${\log }_{8}0.25=-\frac{2}{3}$.

Part b. Step 1. Given.

The given exponential form is $\ln e^4$.

Part b. Step 2. To determine.

We have to evaluate the given expression.

Part b. Step 3. Calculation.

We’ll use the formula: $\ln e^b=b$.

We compare$\ln e^4$ with $\ln e^b$. So, $b=4$.

Hence, $\ln e^4=b=4$.

Part c. Step 1. Given.

The given exponential form is $\ln \left(\frac{1}{e}\right)$.

Part c. Step 2. To determine.

We have to evaluate the given expression.

Part c. Step 3. Calculation.

We’ll use the formula: $\ln e^b=b$.

We compare $\ln \left(\frac{1}{e}\right)=\ln \left(e^{-1}\right)$with $\ln e^b$. So, $b=-1$.

Hence, $\ln \left(\frac{1}{e}\right)=b=-1$.