Question

Evaluating Logarithms Evaluate the expression.

(a) ${\log }_3{3}^7$ (b) ${\log }_{4}64$ (c)${\log }_{5}125$

Short answer

1. The required value is 7.
2. The required value is 3.
3. The required value is 3.

Step-by-step solution

Part a. Step 1. Given.

The given exponential form is ${\log }_3{3}^7$.

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${\log }_3{3}^7$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
${\log }_3{3}^7$	${\log }_3\left(3^7\right)$	3	7	7

Hence,${\log }_3{3}^7$has value role="math" $=7$.

Part b. Step 1. Given.

The given exponential form is ${\log }_{4}64$.

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${\log }_{4}64$ 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
${\log }_{4}64$	${\log }_4\left(4^3\right)$	4	3	3

Hence,${\log }_{4}64$has value $=3$.

Part c. Step 1. Given.

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

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${\log }_{5}125$ 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 $=3$.