Solve For \[$x\$\]:$\[ \log (2x + 1) + \log 5 = \log (x + 6) \\]

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 focus on solving a specific type of logarithmic equation, namely the equation involving the sum of logarithms. We will use the given equation $\log (2x + 1) + \log 5 = \log (x + 6)$ as a case study to illustrate the steps involved in solving such equations.

Understanding the Properties of Logarithms

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithm of a number is the exponent to which a base number must be raised to produce that number. For example, $\log_2 8 = 3$ because $2^3 = 8$. The logarithm of a product is equal to the sum of the logarithms of the individual numbers, and the logarithm of a quotient is equal to the difference of the logarithms of the individual numbers.

The Given Equation

The given equation is $\log (2x + 1) + \log 5 = \log (x + 6)$. Our goal is to solve for $x$.

Step 1: Use the Property of Logarithms to Combine the Terms

Using the property of logarithms that states the logarithm of a product is equal to the sum of the logarithms of the individual numbers, we can rewrite the equation as:

$$\log (2x + 1) + \log 5 = \log (x + 6)$$

$$\log (2x + 1 \cdot 5) = \log (x + 6)$$

$$\log (2x + 5) = \log (x + 6)$$

Step 2: Equate the Arguments of the Logarithms

Since the logarithms on both sides of the equation are equal, we can equate the arguments of the logarithms:

$$2x + 5 = x + 6$$

Step 3: Solve for $x$

Now, we can solve for $x$ by isolating the variable on one side of the equation:

$$2x + 5 = x + 6$$

$$2x - x = 6 - 5$$

$$x = 1$$

Conclusion

In this article, we have solved a logarithmic equation involving the sum of logarithms. We used the properties of logarithms to combine the terms and equate the arguments of the logarithms. Finally, we solved for $x$ by isolating the variable on one side of the equation. This example illustrates the importance of understanding the properties of logarithms in solving logarithmic equations.

Common Mistakes to Avoid

When solving logarithmic equations, it's essential to avoid common mistakes such as:

• Forgetting to use the properties of logarithms to combine the terms
• Not equating the arguments of the logarithms
• Not isolating the variable on one side of the equation

Real-World Applications

Logarithmic equations have numerous real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design electronic circuits and communication systems.

Practice Problems

To practice solving logarithmic equations, try the following problems:

• $\log (3x + 2) + \log 4 = \log (x + 8)$
• $\log (x + 1) + \log 3 = \log (2x + 5)$

Conclusion

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Introduction

In our previous article, we discussed the steps involved in solving logarithmic equations. However, we understand that sometimes, it's not enough to just read about the process; you need to see it in action. That's why we've put together this Q&A guide, where we'll answer some of the most frequently asked questions about solving logarithmic equations.

Q: What is the first step in solving a logarithmic equation?

A: The first step in solving a logarithmic equation is to use the property of logarithms that states the logarithm of a product is equal to the sum of the logarithms of the individual numbers. This allows us to combine the terms on the left-hand side of the equation.

Q: How do I know when to use the property of logarithms to combine the terms?

A: You should use the property of logarithms to combine the terms when the equation involves the sum or product of logarithms. For example, if the equation is $\log (2x + 1) + \log 5 = \log (x + 6)$, you can use the property to combine the terms on the left-hand side.

Q: What is the next step after combining the terms?

A: After combining the terms, the next step is to equate the arguments of the logarithms. This means that you set the expressions inside the logarithms equal to each other.

Q: How do I equate the arguments of the logarithms?

A: To equate the arguments of the logarithms, you simply set the expressions inside the logarithms equal to each other. For example, if the equation is $\log (2x + 5) = \log (x + 6)$, you can equate the arguments by setting $2x + 5 = x + 6$.

Q: What is the final step in solving a logarithmic equation?

A: The final step in solving a logarithmic equation is to solve for $x$. This involves isolating the variable on one side of the equation.

Q: How do I solve for $x$?

A: To solve for $x$, you need to isolate the variable on one side of the equation. This involves using algebraic operations such as addition, subtraction, multiplication, and division to get $x$ by itself.

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 use the property of logarithms to combine the terms
• Not equating the arguments of the logarithms
• Not isolating the variable on one side of the equation

Q: What are some real-world applications of logarithmic equations?

A: Logarithmic equations have numerous real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design electronic circuits and communication systems.

Q: How can I practice solving logarithmic equations?

A: You can practice solving logarithmic equations by trying the following problems:

• $\log (3x + 2) + \log 4 = \log (x + 8)$
• $\log (x + 1) + \log 3 = \log (2x + 5)$

Conclusion

In conclusion, solving logarithmic equations requires a deep understanding of the properties of logarithms. By using the properties of logarithms to combine the terms and equate the arguments of the logarithms, we can solve for $x$ and apply the results to real-world problems. Remember to avoid common mistakes and practice solving logarithmic equations to become proficient in this area of mathematics.

Additional Resources

If you're looking for additional resources to help you practice solving logarithmic equations, we recommend the following:

• Online tutorials: Websites such as Khan Academy and Mathway offer interactive tutorials and practice problems to help you learn and practice solving logarithmic equations.
• Textbooks: There are many textbooks available that cover logarithmic equations, including "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart.
• Practice problems: You can find practice problems online or in textbooks to help you practice solving logarithmic equations.

Final Tips

• Practice regularly: The more you practice solving logarithmic equations, the more comfortable you'll become with the process.
• Use online resources: Websites such as Khan Academy and Mathway offer interactive tutorials and practice problems to help you learn and practice solving logarithmic equations.
• Seek help when needed: If you're struggling with a particular problem or concept, don't be afraid to seek help from a teacher, tutor, or classmate.