Problem 59 - College Algebra

Question

What is the natural exponential function?

Step-by-Step Solution

Answer

The natural exponential function is a mathematical function expressed as \(f(x) = a \cdot e^{x}\), where e is Euler's number, approximately equal to 2.71828, and the base of the natural logarithm. 'a' is a constant and 'x' is a variable.

Step 1: Define exponential function

An exponential function is a mathematical function of the form \(f(x) = a \cdot b^{x}\), where 'a' and 'b' are constants, 'x' is a variable, and 'b' is a positive real number different from 1.

Step 2: Define natural exponential function

When the base 'b' in the exponential function is the mathematical constant e (approximately equal to 2.71828), the function is called the 'natural exponential function'. Therefore, the natural exponential function is of the form \(f(x) = a \cdot e^{x}\).

Step 3: Describe Euler's number

Euler's number (e) is a mathematical constant and is the base of the natural logarithm. It's an irrational number meaning that it cannot be expressed as a simple fraction, and its decimal representation never ends or repeats.