Question

Evaluating Logarithms Evaluate the expression.

(a)${\log }_3\left(\frac{1}{27}\right)$ (b) ${\log }_{10}\sqrt{10}$ (c)${\log }_{5}0.2$

Short answer

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

Step-by-step solution

Part a. Step 1. Given.

The given exponential form is ${\log }_3\left(\frac{1}{27}\right)$.

Part a. Step 2. To determine.

We have to evaluate the given expression.

Part a. Step 3. Calculation.

We’ll use the formula: ${\log }_a\left(a^b\right)=b$.

We will write${\log }_3\left(\frac{1}{27}\right)$ in form${\log }_a\left(a^b\right)$ to find a and b. Then we’ll get the answer.

Given	form	a	b	Answer
${\log }_3\left(\frac{1}{27}\right)$	${\log }_3\left(3^{-3}\right)$	3	-3	-3

Hence,${\log }_3\left(\frac{1}{27}\right)$ has value $=-3$.

Part b. Step 1. Given.

The given exponential form is ${\log }_{10}\sqrt{10}$.

Part b. Step 2. To determine.

We have to evaluate the given expression.

Part b. Step 3. Calculation.

We’ll use the formula: ${\log }_a\left(a^b\right)=b$.

We will write${\log }_{10}\sqrt{10}$ in form${\log }_a\left(a^b\right)$ to find a and b. Then we’ll get the answer.

Given	${\log }_a\left(a^b\right)$form	a	b	Answer
${\log }_{10}\sqrt{10}$	${\log }_{10}\left({10}^{\frac{1}{2}}\right)$	10	$\frac{1}{2}$	$\frac{1}{2}$

Hence,${\log }_{10}\sqrt{10}$ has value $=\frac{1}{2}$.

Part c. Step 1. Given.

The given exponential form is ${\log }_{5}0.2$.

Part c. Step 2. To determine.

We have to evaluate the given expression.

Part c. Step 3. Calculation.

We’ll use the formula: ${\log }_a\left(a^b\right)=b$.

We will write${\log }_{5}0.2$ in form${\log }_a\left(a^b\right)$ to find a and b. Then we’ll get the answer.

Given	${\log }_a\left(a^b\right)$form	a	b	Answer
${\log }_{5}0.2$	${\log }_5\left(5^{-1}\right)$	6	$-1$	$-1$

Hence,${\log }_{5}0.2$ has value $=-1$.