Question

simplify each expression.
$ln e^3 = $
$ln e^{2y} = $

Explanation:

Step1: Recall the logarithm property

The natural logarithm \(\ln\) and the exponential function \(e^x\) are inverse functions, so \(\ln(e^x)=x\) by the property of inverse functions.

Step2: Simplify \(\ln e^3\)

Using the property \(\ln(e^x)=x\), when \(x = 3\), we have \(\ln(e^3)=3\).

Step3: Simplify \(\ln e^{2y}\)

Again, using the property \(\ln(e^x)=x\), when \(x = 2y\), we get \(\ln(e^{2y})=2y\).

Answer:

For \(\ln e^3\), the simplified form is \(3\).
For \(\ln e^{2y}\), the simplified form is \(2y\).