Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in exponential form. \(\ln (w)=n\)

Step-by-Step Solution

Answer

The exponential form is \(e^n = w\).

Step 1: Understand the Natural Logarithm

The natural logarithm function, denoted as \(\ln(x)\), is the inverse of the exponential function \(e^x\). Specifically, \(\ln(w) = n\) means that \(e^n = w\). Our goal is to rewrite this equation in exponential form.

Step 2: Rewrite in Exponential Form

Given \(\ln(w) = n\), apply the property of logarithms that connects logarithmic form to exponential form: \(e^n = w\). This is the exponential form of the given equation.

Key Concepts

Natural Logarithm

The natural logarithm is a logarithm with a special base, known as Euler's number (e), which is approximately 2.71828. It is denoted by the symbol \(\ln(x)\), and is widely used in calculus and complex mathematical equations. Unlike common logarithms that have a base of 10, natural logarithms use \(e\) as their base.
Understanding natural logarithms is fundamental for dealing with exponential growth and decay models, as well as various applications in finance, science, and engineering. The natural logarithm of a number \(x\), written as \(\ln(x)\), tells us what power the base \(e\) must be raised to produce the number \(x\).

• For example, \(\ln(e) = 1\) because \(e^1 = e\).
• Similarly, \(\ln(1) = 0\) because \(e^0 = 1\).

These simple rules highlight why natural logarithms are so important, especially when dealing with exponential functions in mathematics.

Inverse Functions

Inverse functions are functions that reverse the effect of the original function. If you have a function \(f(x)\), its inverse is denoted as \(f^{-1}(x)\). This inverse function basically `undoes' what the original function did.
In the context of natural logarithms, the logarithm function \(\ln(x)\) is the inverse of the exponential function \(e^x\). This means that:

• If you take \(\ln(e^x)\), the result will be \(x\).
• Similarly, if you start with \(x\) and apply \(e\) to it, then find the natural log, such that \(\ln(e^x) = x\).

Understanding inverse functions is crucial in solving mathematical equations as it allows you to isolate variables and find their values. This concept is pivotal in calculus, algebra, and other branches of mathematics when solving complex equations.

Logarithmic Equations

Logarithmic equations are equations that involve logarithms. These equations often require rewriting in a form that is easier to solve or analyze, typically done by converting the logarithmic form into exponential form.
For example, consider the equation \(\ln(w) = n\). The task is to express this in exponential form using the relationship \(y = \ln(x)\) which implies \(e^y = x\). Applying this to the equation, we rewrite \(\ln(w) = n\) as \(e^n = w\).
This conversion from logarithmic to exponential form is not only useful for solving equations but also helps in understanding the growth behavior of various natural phenomena.

• Rewriting logarithmic equations in exponential form simplifies the solution process.
• They are important in applications involving exponential growth or decay, such as population studies or radioactive decay.

Mastering logarithmic equations involves understanding how to manipulate them algebraically to reveal more about the quantities involved.