Common Errors in Logarithmic Equations

Oct 13, 2025 by talynt.org 100 views

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will examine a student solution to a logarithmic equation and identify the error in their approach. We will then provide the correct solution and explain the reasoning behind it.

The Student Solution

The student solution is as follows:

{ \begin{array}{c} 2 \ln (x)=\ln (3 x)-[\ln (9)-2 \ln (3)] \\ \ln \left(x^2\right)=\ln (3 x)-[\ln (9)-\ln (9)] \\ \ln \left(x^2\right)=\ln (3 x)-\ln (9) \\ \ln \left(x^2\right)=\ln \left(\frac{3 x}{9}\right) \\ x^2=\frac{3 x}{9} \\ x^2=\frac{x}{3} \\ x=\pm \sqrt{\frac{x}{3}} \\ x=\pm \frac{1}{\sqrt{3}} x \end{array} }

Identifying the Error

The student solution appears to be correct at first glance, but upon closer inspection, we can see that there is an error in the last step. The student has incorrectly simplified the expression $\frac{3 x}{9}$ to $\frac{x}{3}$ .

Correct Solution

To solve the equation, we need to follow the correct order of operations. We can start by simplifying the expression inside the brackets:

{ \begin{array}{c} 2 \ln (x)=\ln (3 x)-[\ln (9)-2 \ln (3)] \\ \ln \left(x^2\right)=\ln (3 x)-[\ln (9)-\ln (9)] \\ \ln \left(x^2\right)=\ln (3 x)-\ln (9) \\ \ln \left(x^2\right)=\ln \left(\frac{3 x}{9}\right) \\ \ln \left(x^2\right)=\ln \left(\frac{x}{3}\right) \\ x^2=\frac{x}{3} \\ 3 x^2=x \\ x^2=\frac{x}{3} \\ 3 x^2=x \\ x(x-1)=0 \\ x=0 \text{ or } x=1 \end{array} }

Conclusion

In conclusion, the student solution to the logarithmic equation contains an error in the last step. The correct solution involves simplifying the expression inside the brackets and then solving for $x$ . The final answer is $x=0$ or $x=1$ .

Properties of Logarithms

Before we proceed to the next section, let's review some of the properties of logarithms that we used in the solution:

$\ln (a b)=\ln (a)+\ln (b)$
$\ln (a b)=\ln (a)-\ln (b)$
$\ln (a^b)=b \ln (a)$
$\ln (a)-\ln (b)=\ln \left(\frac{a}{b}\right)$

Common Errors in Logarithmic Equations

In this section, we will discuss some common errors that students make when solving logarithmic equations.

Error 1: Incorrect Use of Logarithmic Properties

One common error is the incorrect use of logarithmic properties. For example, students may incorrectly use the property $\ln (a b)=\ln (a)+\ln (b)$ to simplify an expression.

Error 2: Failure to Simplify Expressions

Another common error is the failure to simplify expressions. Students may leave expressions in their original form, rather than simplifying them using logarithmic properties.

Error 3: Incorrect Solution of Equations

Finally, students may incorrectly solve equations. For example, they may incorrectly solve a quadratic equation or fail to check their solutions.

Tips for Solving Logarithmic Equations

Here are some tips for solving logarithmic equations:

Read the problem carefully: Before starting to solve the equation, read it carefully to make sure you understand what is being asked.
Simplify expressions: Simplify expressions using logarithmic properties before solving the equation.
Check your solutions: Check your solutions to make sure they are correct.
Use a calculator: If necessary, use a calculator to check your solutions.

Conclusion

Introduction

Logarithmic equations can be challenging to solve, but with practice and patience, you can become proficient in solving them. In this article, we will provide a Q&A guide to help you understand logarithmic equations and how to solve them.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves logarithms. Logarithms are the inverse of exponents, and they are used to solve equations that involve exponential functions.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

$\ln (a b)=\ln (a)+\ln (b)$
$\ln (a b)=\ln (a)-\ln (b)$
$\ln (a^b)=b \ln (a)$
$\ln (a)-\ln (b)=\ln \left(\frac{a}{b}\right)$

Q: How do I simplify logarithmic expressions?

A: To simplify logarithmic expressions, you can use the properties of logarithms. For example, if you have the expression $\ln (a b)$ , you can simplify it to $\ln (a)+\ln (b)$ .

Q: How do I solve logarithmic equations?

A: To solve logarithmic equations, you need to follow the correct order of operations. First, simplify the expression inside the logarithm. Then, use the properties of logarithms to isolate the variable.

Q: What are some common errors to avoid when solving logarithmic equations?

A: Some common errors to avoid when solving logarithmic equations include:

Incorrectly using logarithmic properties
Failing to simplify expressions
Incorrectly solving equations

Q: How do I check my solutions to logarithmic equations?

A: To check your solutions to logarithmic equations, you can use a calculator to plug in the values and see if they satisfy the equation.

Q: What are some tips for solving logarithmic equations?

A: Some tips for solving logarithmic equations include:

Read the problem carefully before starting to solve the equation
Simplify expressions using logarithmic properties
Check your solutions to make sure they are correct
Use a calculator if necessary

Q: Can you provide an example of a logarithmic equation?

A: Here is an example of a logarithmic equation:

{ \begin{array}{c} 2 \ln (x)=\ln (3 x)-[\ln (9)-2 \ln (3)] \\ \ln \left(x^2\right)=\ln (3 x)-[\ln (9)-\ln (9)] \\ \ln \left(x^2\right)=\ln (3 x)-\ln (9) \\ \ln \left(x^2\right)=\ln \left(\frac{3 x}{9}\right) \\ x^2=\frac{x}{3} \\ 3 x^2=x \\ x(x-1)=0 \\ x=0 \text{ or } x=1 \end{array} }

Q: Can you explain the steps to solve this equation?

A: To solve this equation, we need to follow the correct order of operations. First, simplify the expression inside the logarithm. Then, use the properties of logarithms to isolate the variable.

Here are the steps to solve this equation:

Simplify the expression inside the logarithm: $\ln (3 x)-[\ln (9)-\ln (9)]$
Use the properties of logarithms to simplify the expression: $\ln (3 x)-\ln (9)$
Use the properties of logarithms to isolate the variable: $x^2=\frac{x}{3}$
Solve the equation: $x(x-1)=0$
Check the solutions: $x=0$ or $x=1$

Conclusion

In conclusion, solving logarithmic equations requires a deep understanding of the properties of logarithms and the correct order of operations. By following the steps outlined in this article, you can become proficient in solving logarithmic equations. Remember to read the problem carefully, simplify expressions, check your solutions, and use a calculator if necessary. With practice and patience, you will become proficient in solving logarithmic equations.