To rewrite the given exponential equation [tex]\( 4 e^x = 16 \)[/tex] as a logarithmic equation, follow these steps:
1. Isolate the exponential term:
[tex]\[
e^x = \frac{16}{4}
\][/tex]
Simplify the right side:
[tex]\[
e^x = 4
\][/tex]
2. Take the natural logarithm (ln) of both sides:
[tex]\[
\ln(e^x) = \ln(4)
\][/tex]
3. Apply the logarithmic identity [tex]\(\ln(e^x) = x \cdot \ln(e)\)[/tex]:
[tex]\[
x \cdot \ln(e) = \ln(4)
\][/tex]
4. Since [tex]\(\ln(e)\)[/tex] is 1, the equation simplifies to:
[tex]\[
x = \ln(4)
\][/tex]
Therefore, the logarithmic equation is:
[tex]\[
x = \ln(4)
\][/tex]
