Question

Exponential Form Express the equation in exponential form.

(a) ${\log }_5\left(\frac{1}{5}\right)=-1$ (b)${\log }_{4}64=3$

Short answer

1. Exponential form of ${\log }_5\left(\frac{1}{5}\right)=-1$is $5^{-1}=\frac{1}{5}$.
2. Exponential form of ${\log }_{4}64=3$is $4^3=64$.

Step-by-step solution

Part a. Step 1. Given.

The given logarithmic form is ${\log }_5\left(\frac{1}{5}\right)=-1$.

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

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

Part b. Step 1. Given.

The given logarithmic form is ${\log }_{4}64=3$.

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 ($a^b=c$)
${\log }_{4}64=3$	4	3	64	$4^3=64$

Hence, exponential form of ${\log }_{4}64=3$is $4^3=64$.