Question

Exponential Form Express the equation in exponential form.

(a)$\ln (x+1)=2$ (b)$\ln (x-1)=4$

Short answer

1. Exponential form of $\ln (x+1)=2$is $e^2=x+1$.
2. Exponential form of $\ln (x-1)=4$is $e^4=x-1$.

Step-by-step solution

Part a. Step 1. Given.

The given logarithmic form is $\ln (x+1)=2$.

Part a. Step 2. To determine.

We have to find the exponential form.

Part a. Step 3. Calculation.

We’ll use the formula:$e^b=c$ is equivalent to $\ln c=b$.

Comparing we find the values ofb, c. Then plug them in the exponential form.

Logarithmic form ($\ln c=b$)	b	c	Exponential form ($e^b=c$)
$\ln (x+1)=2$	2	$x+1$	$e^2=x+1$

Hence, exponential form of $\ln (x+1)=2$is $e^2=x+1$.

Part b. Step 1. Given. 

The given logarithmic form is $\ln (x-1)=4$.

Part b. Step 2. To determine.

We have to find the exponential form.

Part b. Step 3. Calculation.

We’ll use the formula:$e^b=c$ is equivalent to $\ln c=b$.

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

Logarithmic form ($\ln c=b$)	b	c	Exponential form ($e^b=c$)
$\ln (x-1)=4$	4	$x-1$	$e^4=x-1$

Hence, exponential form of $\ln (x-1)=4$is $e^4=x-1$.