Solve $\log (4x+5) = 2$. Round To The Nearest Thousandth If Necessary.A. 95 B. 1.978 C. 25 D. 23.75

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 logarithmic equation $\log (4x+5) = 2$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's understand the basics of logarithmic equations. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log_b x$, then $b^y = x$. The logarithm of a number is the power to which a base number must be raised to produce that number.

Solving the Equation

Now that we have a basic understanding of logarithmic equations, let's solve the equation $\log (4x+5) = 2$. To solve this equation, we need to get rid of the logarithm. We can do this by using the definition of a logarithm.

Step 1: Exponentiate Both Sides

The first step in solving the equation is to exponentiate both sides. This means that we will raise the base of the logarithm to the power of the exponent. In this case, the base is 10 (since we are using the common logarithm), and the exponent is 2.

$$\log (4x+5) = 2$$

$$10^{\log (4x+5)} = 10^2$$

Step 2: Simplify the Equation

Now that we have exponentiated both sides, we can simplify the equation. The left-hand side of the equation simplifies to $4x+5$, and the right-hand side simplifies to $100$.

$$4x+5 = 100$$

Step 3: Solve for x

Now that we have a linear equation, we can solve for x. We can do this by subtracting 5 from both sides and then dividing both sides by 4.

$$4x = 95$$

$$x = \frac{95}{4}$$

Step 4: Round to the Nearest Thousandth

Finally, we need to round our answer to the nearest thousandth. This means that we will round our answer to three decimal places.

$$x \approx 23.75$$

Conclusion

In this article, we solved the logarithmic equation $\log (4x+5) = 2$ using the definition of a logarithm. We exponentiated both sides, simplified the equation, and solved for x. Finally, we rounded our answer to the nearest thousandth. The correct answer is $\boxed{23.75}$.

Discussion

Do you have any questions about solving logarithmic equations? Have you encountered any challenging logarithmic equations in the past? Share your thoughts and experiences in the comments below!

Additional Resources

If you are struggling with logarithmic equations, here are some additional resources that may help:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

Final Answer

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Introduction

In our previous article, we solved the logarithmic equation $\log (4x+5) = 2$ using the definition of a logarithm. We exponentiated both sides, simplified the equation, and solved for x. In this article, we will provide a Q&A guide to help you better understand logarithmic equations and how to solve them.

Q&A

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log_b x$, then $b^y = x$. The logarithm of a number is the power to which a base number must be raised to produce that number.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to get rid of the logarithm. You can do this by using the definition of a logarithm, which states that if $y = \log_b x$, then $b^y = x$. You can then exponentiate both sides of the equation and simplify to solve for x.

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

A: A logarithmic equation involves a logarithm, which is the inverse operation of exponentiation. An exponential equation, on the other hand, involves an exponent, which is the power to which a base number must be raised to produce a certain number.

Q: How do I know which base to use when solving a logarithmic equation?

A: The base of a logarithmic equation is usually given in the problem. If it is not given, you can assume that the base is 10 (since we are using the common logarithm).

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

A: Yes, you can use a calculator to solve a logarithmic equation. However, you need to make sure that you are using the correct base and that you are entering the equation correctly.

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 exponentiate both sides of the equation
• Not simplifying the equation correctly
• Not checking the domain of the logarithm
• Not rounding the answer to the correct number of decimal places

Examples

Example 1: Solving a Logarithmic Equation with a Base of 2

Solve the equation $\log_2 (x+1) = 3$.

Step 1: Exponentiate Both Sides

$$\log_2 (x+1) = 3$$

$$2^{\log_2 (x+1)} = 2^3$$

Step 2: Simplify the Equation

$$x+1 = 8$$

Step 3: Solve for x

$$x = 7$$

Example 2: Solving a Logarithmic Equation with a Base of 10

Solve the equation $\log_{10} (x-2) = 2$.

Step 1: Exponentiate Both Sides

$$\log_{10} (x-2) = 2$$

$$10^{\log_{10} (x-2)} = 10^2$$

Step 2: Simplify the Equation

$$x-2 = 100$$

Step 3: Solve for x

$$x = 102$$

Conclusion

In this article, we provided a Q&A guide to help you better understand logarithmic equations and how to solve them. We also provided two examples of solving logarithmic equations with different bases. By following the steps outlined in this article, you should be able to solve logarithmic equations with ease.

Discussion

Do you have any questions about solving logarithmic equations? Have you encountered any challenging logarithmic equations in the past? Share your thoughts and experiences in the comments below!

Additional Resources

If you are struggling with logarithmic equations, here are some additional resources that may help:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

Final Answer

The final answer is: $\boxed{23.75}$