Question

Evaluating Logarithms Evaluate the expression.

(a)${\log }_{6}36$ (b) ${\log }_{9}81$ (c)${\log }_7{7}^{10}$

Short answer

1. The required value is 2.
2. The required value is 2.
3. The required value is 10.

Step-by-step solution

Part a. Step 1. Given.

The given exponential form is ${\log }_{6}36$.

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${\log }_{6}36$ 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
${\log }_{6}36$	${\log }_6\left(6^2\right)$	6	2	2

Hence,${\log }_{6}36$ has value $=2$.

Part b. Step 1. Given.

The given exponential form is ${\log }_{9}81$.

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${\log }_{9}81$ 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
${\log }_{9}81$	${\log }_9\left(9^2\right)$	9	2	2

Hence,${\log }_{9}81$ has value $=2$.

Part c. Step 1. Given.

The given exponential form is ${\log }_7{7}^{10}$.

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${\log }_7{7}^{10}$ 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
${\log }_7{7}^{10}$	${\log }_7\left(7^{10}\right)$	7	10	10

Hence,${\log }_7{7}^{10}$ has value $=10$.