Solve The Equation:$ \log_9(x-2) + \log_9 4 = 1 $

Feb 26, 2025 by ADMIN 50 views

**Solving Logarithmic Equations: A Step-by-Step Guide**

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Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $\log_9(x-2) + \log_9 4 = 1$ . This equation involves logarithms with the same base, and we will use properties of logarithms to simplify and solve it.

Understanding Logarithms

Before we dive into solving the equation, let's quickly review the basics of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$ , then $\log_a c = b$ . The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number.

For example, $\log_2 8 = 3$ because $2^3 = 8$ . Similarly, $\log_9 9 = 1$ because $9^1 = 9$ .

Properties of Logarithms

There are several properties of logarithms that we will use to simplify and solve the equation. These properties are:

Product Property: $\log_a (xy) = \log_a x + \log_a y$
Quotient Property: $\log_a \frac{x}{y} = \log_a x - \log_a y$
Power Property: $\log_a x^y = y \log_a x$

Solving the Equation

Now that we have reviewed the basics of logarithms and their properties, let's focus on solving the equation $\log_9(x-2) + \log_9 4 = 1$ .

Step 1: Combine the Logarithms

Using the product property of logarithms, we can combine the two logarithms on the left-hand side of the equation:

\log_9(x-2) + \log_9 4 = \log_9((x-2) \cdot 4)

This simplifies to:

\log_9(x-2) + \log_9 4 = \log_9(4x-8)

Step 2: Equate the Arguments

Since the logarithms have the same base, we can equate the arguments of the logarithms:

4x-8 = 9^1

This simplifies to:

4x-8 = 9

Step 3: Solve for x

Now that we have a linear equation, we can solve for x:

4x = 9 + 8

4x = 17

x = \frac{17}{4}

Conclusion

Solving logarithmic equations requires a good understanding of the properties of logarithms. By combining the logarithms, equating the arguments, and solving for x, we can solve equations involving logarithms with the same base. In this article, we solved the equation $\log_9(x-2) + \log_9 4 = 1$ using these steps.

Example Problems

Here are a few example problems to help you practice solving logarithmic equations:

$\log_2(x+1) + \log_2 3 = 2$
$\log_5(x-1) - \log_5 2 = 1$
$\log_3(x+2) + \log_3 4 = 2$

Tips and Tricks

Here are a few tips and tricks to help you solve logarithmic equations:

Always start by combining the logarithms using the product or quotient property.
Equate the arguments of the logarithms and simplify the equation.
Solve for x using algebraic techniques.
Check your answer by plugging it back into the original equation.

By following these steps and tips, you can solve logarithmic equations with ease.

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Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on answering frequently asked questions about logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$ , then $\log_a c = b$ .

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to follow these steps:

Combine the logarithms using the product or quotient property.
Equate the arguments of the logarithms and simplify the equation.
Solve for x using algebraic techniques.
Check your answer by plugging it back into the original equation.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation involves a logarithm, while an exponential equation involves an exponent. For example, $\log_2 x = 3$ is a logarithmic equation, while $2^x = 8$ is an exponential equation.

Q: How do I deal with logarithms with different bases?

A: When dealing with logarithms with different bases, you need to use the change of base formula. The change of base formula is:

\log_a b = \frac{\log_c b}{\log_c a}

where $c$ is any positive number not equal to 1.

Q: Can I use a calculator to solve logarithmic equations?

A: Yes, you can use a calculator to solve logarithmic equations. However, you need to make sure that the calculator is set to the correct base and that you are using the correct operation.

Q: How do I check my answer to a logarithmic equation?

A: To check your answer to a logarithmic equation, you need to plug it back into the original equation and simplify. If the equation is true, then your answer is correct.

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

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

Forgetting to combine the logarithms using the product or quotient property.
Equating the arguments of the logarithms incorrectly.
Solving for x using algebraic techniques incorrectly.
Not checking the answer by plugging it back into the original equation.

Q: Can I use logarithmic equations to solve real-world problems?

A: Yes, you can use logarithmic equations to solve real-world problems. For example, you can use logarithmic equations to model population growth, chemical reactions, and financial transactions.

Q: How do I apply logarithmic equations to real-world problems?

A: To apply logarithmic equations to real-world problems, you need to follow these steps:

Identify the problem and the variables involved.
Write an equation that models the problem.
Solve the equation using logarithmic techniques.
Interpret the results and make conclusions.

By following these steps and tips, you can apply logarithmic equations to real-world problems and solve them with ease.

Conclusion

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. By following the steps and tips outlined in this article, you can solve logarithmic equations and apply them to real-world problems. Remember to combine the logarithms, equate the arguments, solve for x, and check your answer to ensure that you are getting the correct solution.