Which Of The Following Shows The True Solution To The Equation? Log ⁡ ( X ) + Log ⁡ ( X + 5 ) = Log ⁡ ( 6 X + 12 \log(x) + \log(x+5) = \log(6x+12 Lo G ( X ) + Lo G ( X + 5 ) = Lo G ( 6 X + 12 ]A. { X = -3$}$ B. { X = 4$}$ C. { X = -3$}$ And { X = 4$}$ D. { X = -3$}$ And [$x =

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Introduction

In this article, we will delve into the world of logarithmic equations and explore the solution to the equation log⁡(x)+log⁡(x+5)=log⁡(6x+12)\log(x) + \log(x+5) = \log(6x+12). We will examine each option carefully and determine which one shows the true solution to the equation.

Understanding Logarithmic Equations

Before we dive into the solution, let's take a moment to understand 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⁡(x)y = \log(x), then x=10yx = 10^y (assuming we are working with base 10 logarithms).

The Given Equation

The given equation is log⁡(x)+log⁡(x+5)=log⁡(6x+12)\log(x) + \log(x+5) = \log(6x+12). Our goal is to solve for xx.

Step 1: Combine the Logarithms on the Left Side

Using the property of logarithms that states log⁡(a)+log⁡(b)=log⁡(ab)\log(a) + \log(b) = \log(ab), we can combine the logarithms on the left side of the equation:

log⁡(x)+log⁡(x+5)=log⁡(x(x+5))=log⁡(x2+5x)\log(x) + \log(x+5) = \log(x(x+5)) = \log(x^2 + 5x)

Step 2: Equate the Logarithms

Now that we have combined the logarithms on the left side, we can equate the two sides of the equation:

log⁡(x2+5x)=log⁡(6x+12)\log(x^2 + 5x) = \log(6x+12)

Step 3: Eliminate the Logarithms

Since the logarithms on both sides of the equation are equal, we can eliminate the logarithms by exponentiating both sides of the equation:

x2+5x=6x+12x^2 + 5x = 6x + 12

Step 4: Solve for x

Now that we have eliminated the logarithms, we can solve for xx:

x2−x−12=0x^2 - x - 12 = 0

We can factor the quadratic equation as:

(x−4)(x+3)=0(x - 4)(x + 3) = 0

This gives us two possible solutions for xx:

x=4 or x=−3x = 4 \text{ or } x = -3

Analyzing the Options

Now that we have found the solutions to the equation, let's analyze the options:

A. x=−3x = -3 B. x=4x = 4 C. x=−3x = -3 and x=4x = 4 D. x=−3x = -3 and x=4x = 4

Conclusion

Based on our analysis, we can see that options A, B, C, and D all contain one or both of the solutions to the equation. However, the correct answer is not just one of the solutions, but rather both solutions.

Therefore, the correct answer is:

C. x=−3x = -3 and x=4x = 4

This is because both x=−3x = -3 and x=4x = 4 are solutions to the equation, and option C is the only option that includes both solutions.

Final Thoughts

In conclusion, solving logarithmic equations requires a step-by-step approach, including combining logarithms, equating logarithms, eliminating logarithms, and solving for the variable. By following these steps, we can find the solutions to the equation and determine the correct answer.

Additional Resources

For more information on logarithmic equations and how to solve them, check out the following resources:

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

References

  • "Logarithmic Equations" by Khan Academy
  • "Logarithmic Equations" by Mathway
  • "Logarithmic Equations" by Wolfram Alpha
    Logarithmic Equations Q&A ==========================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about logarithmic equations.

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⁡(x)y = \log(x), then x=10yx = 10^y (assuming we are working with base 10 logarithms).

Q: How do I solve a logarithmic equation?

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

  1. Combine the logarithms on the left side of the equation using the property of logarithms that states log⁡(a)+log⁡(b)=log⁡(ab)\log(a) + \log(b) = \log(ab).
  2. Equate the logarithms on both sides of the equation.
  3. Eliminate the logarithms by exponentiating both sides of the equation.
  4. Solve for the variable.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation log⁡(x)=2\log(x) = 2 is a logarithmic equation, while the equation x2=4x^2 = 4 is an exponential equation.

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

A: Yes, you can use a calculator to solve a logarithmic equation. However, it's always a good idea to check your work by plugging the solution back into the original equation.

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

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

  • Not combining the logarithms on the left side of the equation.
  • Not equating the logarithms on both sides of the equation.
  • Not eliminating the logarithms by exponentiating both sides of the equation.
  • Not checking the solution by plugging it back into the original equation.

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

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

Q: What are some common applications of logarithmic equations?

A: Some common applications of logarithmic equations include:

  • Modeling population growth and decline.
  • Calculating the pH of a solution.
  • Determining the amount of time it takes for a chemical reaction to occur.
  • Calculating the interest on a loan or investment.

Q: Can I use logarithmic equations to solve systems of equations?

A: Yes, you can use logarithmic equations to solve systems of equations. However, it's often easier to use other methods, such as substitution or elimination, to solve systems of equations.

Q: What are some tips for solving logarithmic equations?

A: Some tips for solving logarithmic equations include:

  • Start by combining the logarithms on the left side of the equation.
  • Use the property of logarithms that states log⁡(a)+log⁡(b)=log⁡(ab)\log(a) + \log(b) = \log(ab) to simplify the equation.
  • Eliminate the logarithms by exponentiating both sides of the equation.
  • Check the solution by plugging it back into the original equation.

Conclusion

In conclusion, logarithmic equations are a powerful tool for solving a wide range of problems. By following the steps outlined in this article, you can solve logarithmic equations and apply them to real-world problems.

Additional Resources

For more information on logarithmic equations and how to solve them, check out the following resources:

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

References

  • "Logarithmic Equations" by Khan Academy
  • "Logarithmic Equations" by Mathway
  • "Logarithmic Equations" by Wolfram Alpha