Question

evaluate or simplify the expression without using a calculator.
$e^{\ln 3x^5}$
$e^{\ln 3x^5} = \square$

Explanation:

Step1: Recall the inverse property of exponential and logarithmic functions

The exponential function \( e^x \) and the natural logarithmic function \( \ln x \) are inverse functions of each other. The property that describes this relationship is \( e^{\ln a}=a \) for any positive real number \( a \).

Step2: Apply the inverse property to the given expression

In the expression \( e^{\ln 3x^{5}} \), we can identify \( a = 3x^{5} \) (assuming \( 3x^{5}>0 \), which is true for the domain where the natural logarithm is defined). Using the property \( e^{\ln a}=a \), we substitute \( a = 3x^{5} \) into the property. So \( e^{\ln 3x^{5}}=3x^{5} \).

Answer:

\( 3x^{5} \)