Sure! Let's rewrite the given exponential equation [tex]\( 4e^x = 16 \)[/tex] as a logarithmic equation. Follow these steps:
### Step 1: Isolate [tex]\( e^x \)[/tex]
First, divide both sides of the equation by 4 to isolate [tex]\( e^x \)[/tex]:
[tex]\[
e^x = \frac{16}{4}
\][/tex]
Simplify the fraction:
[tex]\[
e^x = 4
\][/tex]
### Step 2: Apply the Natural Logarithm
To solve for [tex]\( x \)[/tex], apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the inverse operation of the exponential function with base [tex]\( e \)[/tex]:
[tex]\[
\ln(e^x) = \ln(4)
\][/tex]
### Step 3: Simplify Using Logarithm Properties
Using the property of logarithms that [tex]\( \ln(e^x) = x \)[/tex]:
[tex]\[
x = \ln(4)
\][/tex]
So, the logarithmic form of the given equation is:
[tex]\[
x = \ln(4)
\][/tex]
Therefore, the correct answer is:
[tex]\[
x = \ln(4) \approx 1.3862943611198906
\][/tex]
