Problem 78 - College Algebra

Question

What is the natural exponential function?

Step-by-Step Solution

Answer

The natural exponential function is expressed as \( e^x \), where \( e \) is the Euler's number. It exhibits properties like being its own derivative, always positive and increasing, and able to multiply similar base exponentials together.

Step 1: Definition

The natural exponential function is denoted as \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. This function is defined for all real numbers, and it's always positive.

Step 2: Properties

Some remarkable properties of natural exponential functions include derivative and integral of \( e^x \) is \( e^x \) itself, \( e^x \) is always increasing and never equals to zero. Moreover, the sum of exponents allows similar base exponentials to multiply together, i.e., \( e^{x}*e^{y} = e^{x+y} \).