Question

evaluate or simplify the expression without using a calculator.\\( e^{ln 2x^4} \\)\\( e^{ln 2x^4} = \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 \( e^{\ln 2x^{4}} \), we can identify \( a = 2x^{4} \) (assuming \( 2x^{4}>0 \), which is true for all real numbers \( x
eq0 \), and if \( x = 0 \), the original expression \( \ln(2x^{4}) \) is not defined, so we consider the domain where \( x
eq0 \)). Using the property \( e^{\ln a}=a \), we substitute \( a = 2x^{4} \) into the property. So \( e^{\ln 2x^{4}}=2x^{4} \).

Answer:

\( 2x^{4} \)