Problem 85 - Precalculus

Question

Explain the mistake that is made. Solve the equation: \(4 e^{x}=9\) Solution: Take the natural log of both sides. \(\quad \ln \left(4 e^{x}\right)=\ln 9\) Apply the property of inverses. \(4 x=\ln 9\) \(x=\frac{\ln 9}{4} \approx 0.55\) Solve for \(x\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was not correctly applying logarithm properties; the correct solution is \(x = \ln\left(\frac{9}{4}\right) \approx 0.8109\).

Step 1: Identify the equation

First, we observe the given equation: \(4 e^{x} = 9\). Our goal is to solve for \(x\).

Step 2: Apply natural logarithm

Take the natural logarithm of both sides to solve for \(x\) in terms of natural logs: \[\ln(4 e^x) = \ln 9\]

Step 3: Simplify the equation

Use the property of logarithms, \(\ln(ab) = \ln a + \ln b\), to separate the expression:\[\ln 4 + \ln e^x = \ln 9\]

Step 4: Apply the property of inverses

Recognize that \(\ln e^x = x\). Substitute this into the equation:\[\ln 4 + x = \ln 9\]

Step 5: Solve for x

Isolate \(x\) on one side:\[x = \ln 9 - \ln 4\]Convert the difference of logs to the division of arguments, using the property \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\):\[x = \ln\left(\frac{9}{4}\right)\]

Step 6: Calculate x

Find the numerical value:\[ x \approx \ln\left(\frac{9}{4}\right) \approx 0.8109 \]

Step 7: Conclusion: Identify the error

The mistake in the original solution was incorrectly assuming \(\ln(4 e^x) = 4x\). The correct property of logarithms involves separating \(\ln(4) + \ln(e^x)\).

Key Concepts

Natural Logarithms

Natural logarithms are a powerful tool in mathematics for solving exponential equations. A natural logarithm uses the base \( e \), which is approximately 2.718. This special number, \( e \), is used widely across different fields such as calculus, finance, and biology.
Understanding natural logarithms can simplify complex exponential expressions. When dealing with equations involving \( e^x \), taking the natural logarithm of both sides allows us to manipulate and solve for \( x \).
For instance, in the equation \( 4 e^{x} = 9 \), applying \( \ln \) on both sides gives us \( \ln(4 e^x) = \ln 9 \). This step makes it possible to break down the complexities of the equation into simpler components, leading to a solution for \( x \).
Remember that:

• Natural logarithms convert exponential equations into linear forms, making it easier to find unknown variables.
• They are specifically useful when the equation involves the number \( e \), because \( \ln(e^x) = x \).

Logarithmic Properties

Logarithmic properties are essential when solving equations involving exponents. Understanding these properties enables you to simplify and solve exponential equations accurately without common mistakes that lead to errors.
Some key properties include:

• Product Property: \( \ln(ab) = \ln a + \ln b \). This allows you to separate the components of a product into a sum of logarithms.
• Quotient Property: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \). This is useful for simplifying expressions where division is involved.
• Power Property: \( \ln(a^n) = n \ln a \). This is applied when handling powers within a logarithmic expression.

In our solved equation, the product property helps us to split \( \ln(4 e^x) \) into \( \ln 4 + \ln e^x \). Understanding and applying these properties correctly simplifies the equation into a form that is easier to solve.
By expertly using these logarithmic properties, you can avoid pitfalls and find accurate solutions for complex equations.

Common Mistakes in Algebra

In algebra, solving equations often involves multiple steps and applying various rules correctly. However, common mistakes can lead to incorrect results if you're not careful.
One such mistake in this problem was assuming \( \ln(4 e^x) = 4x \). This error arose from misunderstanding how natural logarithms apply to products. Instead, the correct application should separate the terms using the product property, resulting in \( \ln 4 + \ln e^x \).
Here are common mistakes and tips to avoid them:

• Misapplying Logarithmic Rules: Always verify which property should be used. For example, use the product property instead of inadvertently simplifying incorrectly, as seen in this case.
• Skipping Steps: Ensure you write down each step clearly. This reduces errors and allows you to backtrack if needed.
• Wrong Operations: Double-check operations, particularly when dealing with fractions or combined expressions.

Accuracy in algebra involves understanding the fundamentals deeply, practicing regularly, and reflecting on mistakes to learn from them. This approach helps you become proficient and precise in solving equations.