Question

Exponential Form Express the equation in exponential form.

(a) ${\log }_5\left(\frac{1}{125}\right)=-3$ (b)${\log }_{8}4=\frac{2}{3}$

Short answer

1. Exponential form of${\log }_5\left(\frac{1}{125}\right)=-3$ is $5^{-3}=\frac{1}{125}$.
2. Exponential form of ${\log }_{8}4=\frac{2}{3}$is $8^{\frac{2}{3}}=4$.

Step-by-step solution

Part a. Step 1. Given.

The given logarithmic form is ${\log }_5\left(\frac{1}{125}\right)=-3$.

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 ($a^b=c$)
${\log }_5\left(\frac{1}{125}\right)=-3$	5	$-3$	$\frac{1}{125}$	$5^{-3}=\frac{1}{125}$

Hence, exponential form of ${\log }_5\left(\frac{1}{125}\right)=-3$is $5^{-3}=\frac{1}{125}$.

Part b. Step 1. Given.

The given logarithmic form is ${\log }_{8}4=\frac{2}{3}$.

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 ($a^b=c$)
${\log }_{8}4=\frac{2}{3}$	8	$\frac{2}{3}$	4	$8^{\frac{2}{3}}=4$

Hence, exponential form of ${\log }_{8}4=\frac{2}{3}$is $8^{\frac{2}{3}}=4$.