Question

Exponential Form Express the equation in exponential form.

(a) ${\log }_{8}2=\frac{1}{3}$ (b)${\log }_{10}0.01=-2$

Short answer

1. Exponential form of ${\log }_{8}2=\frac{1}{3}$is $8^{\frac{1}{3}}=2$.
2. Exponential form of ${\log }_{10}0.01=-2$is ${10}^{-2}=0.01$.

Step-by-step solution

Part a. Step 1. Given.

The given logarithmic form is ${\log }_{8}2=\frac{1}{3}$.

Part a. Step 2. To determine.

We have to find the exponential form.

Part a. Step 3. Calculation.

We’ll use the formula:$a^b=c$ is equivalent to ${\log }_{a}c=b$.

Comparing we find the values of a, b, c. Then plug them in the exponential form.

Logarithmic form (${\log }_{a}c=b$)	a	b	c	Exponential form ($a^b=c$)
${\log }_{8}2=\frac{1}{3}$	8	$\frac{1}{3}$	2	$8^{\frac{1}{3}}=2$

Hence, exponential form of ${\log }_{8}2=\frac{1}{3}$is $8^{\frac{1}{3}}=2$.

Part b. Step 1. Given.

The given logarithmic form is ${\log }_{10}0.01=-2$.

Part b. Step 2. To determine.

We have to find the exponential form.

Part b. Step 3. Calculation.

We’ll use the formula:$a^b=c$ is equivalent to ${\log }_{a}c=b$.

Comparing we find the values of a, b, c. Then plug them in the exponential form.

Logarithmic form (${\log }_{a}c=b$)	a	b	c	Exponential form ($a^b=c$)
${\log }_{10}0.01=-2$	10	$-2$	0.01	${10}^{-2}=0.01$

Hence, exponential form of ${\log }_{10}0.01=-2$is ${10}^{-2}=0.01$.