Question

Logarithmic Form Express the equation in logarithmic form.

(a)$3^{2x}=10$ (b)${10}^{-4x}=0.1$

Short answer

1. Logarithmic form of $3^{2x}=10$is ${\log }_2\left(10\right)=2x$.
2. Logarithmic form of ${10}^{-4x}=0.1$is ${\log }_{10}\left(0.1\right)=-4x$.

Step-by-step solution

Part a. Step 1. Given.

The given exponential form is $3^{2x}=10$.

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.

Exponential form ($a^b=c$)	a	b	c	Logarithmic form (${\log }_{a}c=b$)
$3^{2x}=10$	3	2x	10	${\log }_2\left(10\right)=2x$

Hence, logarithmic form of $3^{2x}=10$is ${\log }_2\left(10\right)=2x$.

Part b. Step 1. Given.

The given exponential form is ${10}^{-4x}=0.1$.

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.

Exponential form ($a^b=c$)	a	b	c	Logarithmic form (${\log }_{a}c=b$)
${10}^{-4x}=0.1$	10	$-4x$	0.1	${\log }_{10}\left(0.1\right)=-4x$

Hence, logarithmic form of ${10}^{-4x}=0.1$is ${\log }_{10}\left(0.1\right)=-4x$.