Question

Logarithmic Form Express the equation in logarithmic form.

(a)$e^{x+1}=0.5$ (b)$e^{0.5x}=t$

Short answer

1. Logarithmic form of $e^{x+1}=0.5$is $\ln \left(0.5\right)=x+1$.
2. Logarithmic form of $e^{0.5x}=t$is $\ln \left(t\right)=0.5x$.

Step-by-step solution

Part a. Step 1. Given.

The given exponential form is $e^{x+1}=0.5$.

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 of b, c. Then plug them in the exponential form.

Exponential form ($e^b=c$)	b	c	Logarithmic form ($\ln c=b$)
$e^{x+1}=0.5$	$x+1$	0.5	$\ln \left(0.5\right)=x+1$

Hence, logarithmic form of $e^{x+1}=0.5$is $\ln \left(0.5\right)=x+1$.

Part b. Step 1. Given.

The given exponential form is $e^{0.5x}=t$.

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.

Exponential form ($e^b=c$)	b	c	Logarithmic form ($\ln c=b$)
$e^{0.5x}=t$	$0.5x$	t	$\ln \left(t\right)=0.5x$

Hence, logarithmic form of $e^{0.5x}=t$is $\ln \left(t\right)=0.5x$.