Question

Evaluating Logarithms Evaluate the expression.

(a)${\log }_{5}125$ (b) ${\log }_{49}7$ (c)${\log }_9\sqrt{3}$

Short answer

1. The required value is 3.
2. The required value is$\frac{1}{2}$.
3. The required value is $\frac{1}{4}$.

Step-by-step solution

Part a. Step 1. Given.

The given exponential form is ${\log }_{5}125$.

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${\log }_{5}125$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
${\log }_{5}125$	${\log }_5\left(5^3\right)$	5	3	3

Hence,${\log }_{5}125$has value $=3$.

Part b. Step 1. Given.

The given exponential form is ${\log }_{49}7$.

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${\log }_{49}7$ 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
${\log }_{49}7$	${\log }_{49}\left(7^{\frac{1}{2}}\right)$	7	$\frac{1}{2}$	$\frac{1}{2}$

Hence,${\log }_{49}7$ has value $=\frac{1}{2}$.

Part c. Step 1. Given.

The given exponential form is ${\log }_9\sqrt{3}$.

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${\log }_9\sqrt{3}$ 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
${\log }_9\sqrt{3}$	${\log }_9\left(9^{\frac{1}{4}}\right)$	9	$\frac{1}{4}$	$\frac{1}{4}$

Hence,${\log }_9\sqrt{3}$ has value $=\frac{1}{4}$.