Question

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

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. By the inverse property of functions, for any positive real number \( a \), we have \( e^{\ln a}=a \).

Step2: Apply the inverse property to the given expression

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

Answer:

\( 3x^{3} \)