Problem 58 - College Algebra

Question

Recall that an exponential function is any equation written in the form \(f(x)=a \cdot b^{x}\) such that \(a\) and \(b\) are positive numbers and \(b \neq 1\) . Any positive number \(b\) can be written as \(b=e^{n}\) for some value of \(n .\) Use this fact to rewrite the formula for an exponential function that uses the number \(e\) as a base.

Step-by-Step Solution

Answer

Rewrite as \(f(x) = a \cdot e^{nx}\).

Step 1: Understand the General Exponential Function

An exponential function is generally written as \(f(x) = a \cdot b^{x}\), where \(a\) and \(b\) are positive constants, and \(b eq 1\). Our goal is to express this using the natural exponential base \(e\).

Step 2: Express b as a Power of e

Since any positive number \(b\) can be expressed as \(b = e^{n}\) for some number \(n\), we substitute this into our exponential function formula.

Step 3: Substitute and Rearrange the Function

Replace \(b\) in the formula \(f(x) = a \cdot b^{x}\) with \(e^{n}\). Thus, the function becomes \(f(x) = a \cdot (e^{n})^{x}\).

Step 4: Simplify Using Exponent Rules

Using the exponent rule \((e^{n})^{x} = e^{nx}\), simplify the function to \(f(x) = a \cdot e^{nx}\). This presents the exponential function using \(e\) as the base.

Key Concepts

Exponential Equation

An exponential equation is a type of mathematical expression where a variable appears in the exponent. These equations take the specific form of:

• \(f(x) = a \cdot b^{x}\)

In this equation, "a" is a coefficient that stretches or compresses the graph vertically, while "b" is the base, responsible for the growth rate of the function. It's essential that "b" is greater than zero and not equal to one, which ensures the equation truly represents exponential growth or decay rather than a linear change.

The variable, often denoted as "x," is the exponent and controls the degree of growth or decay, creating a curve that rapidly increases or decreases. When graphed, exponential equations produce characteristic J-shaped curves.

By understanding exponential equations, we can better comprehend processes in natural sciences and finance, where such growth and decay frequently appear. Common examples include population growth, compound interest calculations, and radioactive decay.

Natural Exponential Base

The natural exponential base, \(e\), is a mathematical constant approximately equal to 2.71828. It is often referred to as Euler’s number and plays a crucial role in describing exponential growth and decay.

What makes \(e\) unique is its natural appearance in many natural phenomena, and it serves as the base for natural logarithms. Unlike other bases, the use of \(e\) simplifies differentiation and integration in calculus, making it a popular choice for mathematicians and scientists.

• In terms of equations, using the natural exponential base allows for smooth and continuous growth rates.
• It appears in various mathematical contexts beyond just exponential functions, including growth processes in biology and chemistry, and financial calculations involving continuous compounding.

To express an exponential function using the base \(e\), we rewrite the base "b" in terms of \(e\) as: \(b = e^{n}\). This substitution helps introduce the natural exponential base into the function, resulting in the simplified form \(f(x) = a \cdot e^{nx}\).

Exponent Rules

Exponent rules are a set of mathematical guidelines that help simplify the operations of exponential expressions. These rules are applicable in simplifying functions that involve powers or exponents.

Here are some fundamental exponent rules that are crucial when working with exponential functions:

• Product of Powers: \(a^{m} \cdot a^{n} = a^{m+n}\) — Multiply powers with the same base by adding their exponents.
• Power of a Power: \((a^{m})^{n} = a^{m \cdot n}\) — Raise a power to a power by multiplying the exponents.
• Quotient of Powers: \(a^{m} / a^{n} = a^{m-n}\) — Divide powers with the same base by subtracting the exponents.

Understanding these rules allows one to rewrite and simplify exponential functions effectively. In the exercise, we leveraged the power of a power rule \((e^{n})^{x} = e^{nx}\) to express the exponential function with \(e\) as the base. This simplification is particularly useful when converting general exponential functions into natural exponential forms, facilitating easier manipulation and analysis of the function.