Problem 61 - Algebra 2

Question

Give an example of an exponential equation that requires using natural logarithms instead of common logarithms to solve.

Step-by-Step Solution

Answer

Solve the equation \( e^x = 5 \) by taking the natural logarithm, resulting in \( x = \ln(5) \approx 1.61 \).

Step 1: Choose an Exponential Equation

Let's consider the exponential equation \( e^x = 5 \). This equation involves the natural exponential constant \( e \) and will require the use of natural logarithms for its solution.

Step 2: Apply the Natural Logarithm

To solve for \( x \), take the natural logarithm (\( \ln \)) of both sides of the equation. This will allow us to use the property \( \ln(e^x) = x \). So, we have:\[ \ln(e^x) = \ln(5) \]

Step 3: Simplify the Equation Using Logarithm Rules

Applying the property of logarithms, \( \ln(e^x) = x \cdot \ln(e) \). Since \( \ln(e) = 1 \), the equation simplifies to:\[ x = \ln(5) \]

Step 4: Calculate the Value of x

Finally, calculate \( x \) using a calculator. We find:\[ x \approx 1.61 \]

Key Concepts

Understanding Exponential Equations

An exponential equation is an equation in which the variable, usually denoted by x, appears in the exponent. These equations have the form \( a^x = b \), where \( a \) and \( b \) are constants. Exponential equations often appear in real-world applications such as population growth, radioactive decay, and compound interest.

For instance, in the equation \( e^x = 5 \), \( e \) is the base, which is a mathematical constant approximately equal to 2.718. The unique aspect of exponential equations is the variable in the exponent, requiring specific methods to solve for it. Unlike linear equations, exponential equations grow much faster and need logarithms for isolation of the variable.

In problems involving exponential equations with base \( e \), we typically use natural logarithms for solving them effectively.

Properties of Logarithms

Logarithms are the inverse functions of exponential functions, making them vital tools for solving exponential equations. Understanding the properties of logarithms allows us to manipulate and solve equations more easily.

Some key properties include:

• \( \log_b(a \ imes c) = \log_b(a) + \log_b(c) \)
• \( \log_b(a/c) = \log_b(a) - \log_b(c) \)
• \( \log_b(a^n) = n \cdot \log_b(a) \)
• \( \log_b(b) = 1 \)

In the context of natural logarithms, when our base is the constant \( e \), we use \( \ln \) to represent logarithms. This simplifies many calculations since \( \ln(e^x) \) reduces to \( x \), due to the fact that \( \ln(e) = 1 \).

These properties allow us to transform and solve exponential equations, shifting the variable from an exponent to being a coefficient, making these equations much more manageable.

Solving Exponential Equations

Solving exponential equations often involves using logarithms to isolate the variable in the exponent. Let's solve the equation \( e^x = 5 \) as an example further illustrating this process.

First, recognize that the equation has the base \( e \). Therefore, we use natural logarithms to solve it. We start by taking the natural logarithm of both sides:

\[ \ln(e^x) = \ln(5) \]

Using the property \( \ln(e^x) = x \cdot \ln(e) \) and knowing that \( \ln(e) = 1 \), the equation simplifies to:

\[ x = \ln(5) \]

This transforms the exponential equation into a simple arithmetic solution. Finally, use a calculator to find \( \ln(5) \), which approximates to 1.61. Thus, \( x \approx 1.61 \).

This method showcases the power of logarithms in tackling equations that don't have immediate arithmetic solutions.

Significance of Constants in Mathematics

Constants in mathematics are fixed values that do not change and are fundamental in forming mathematical expressions and equations. In the context of exponential equations, \( e \), approximately equal to 2.718, is a crucial constant known as the natural logarithm base.

\( e \) appears in a variety of mathematical fields, demonstrating its versatility:

• Growth and decay models (e.g., population dynamics)
• Financial calculations (e.g., continuous compounding interest)
• Complex number calculations (e.g., Euler's formula)

The importance of constants such as \( e \) extends beyond simple calculations. They represent the foundation of many natural and theoretical processes, making them indispensable in both academic and real-world applications.

Developing a strong understanding of these constants and their properties helps pave the way for solving more complex problems in mathematics efficiently.