Solve For $x$:$\log _5(5x-4) - \log _5(-2x+16) = \log _5\left(\frac{1}{x}\right)$

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Introduction

Logarithmic equations can be challenging to solve, especially when they involve multiple logarithms with different bases. In this article, we will focus on solving a logarithmic equation that involves the subtraction of two logarithms with the same base and the addition of a logarithm with a different base. We will use properties of logarithms to simplify the equation and solve for the variable x.

Understanding the Equation

The given equation is:

$$\log _5(5x-4) - \log _5(-2x+16) = \log _5\left(\frac{1}{x}\right)$$

This equation involves three logarithms with base 5. The first two logarithms are subtracted, and the result is equal to the third logarithm. To solve this equation, we need to use properties of logarithms to simplify it.

Using Properties of Logarithms

One of the properties of logarithms is that the subtraction of two logarithms with the same base is equal to the logarithm of the quotient of the two arguments. This property can be written as:

$$\log _b(x) - \log _b(y) = \log _b\left(\frac{x}{y}\right)$$

Using this property, we can rewrite the first two logarithms in the given equation as a single logarithm:

$$\log _5(5x-4) - \log _5(-2x+16) = \log _5\left(\frac{5x-4}{-2x+16}\right)$$

Now, the equation becomes:

$$\log _5\left(\frac{5x-4}{-2x+16}\right) = \log _5\left(\frac{1}{x}\right)$$

Equating the Arguments

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

$$\frac{5x-4}{-2x+16} = \frac{1}{x}$$

Solving for x

To solve for x, we can start by cross-multiplying:

$$(5x-4)x = -2x+16$$

Expanding the left-hand side, we get:

$$5x^2 - 4x = -2x + 16$$

Moving all terms to the left-hand side, we get:

$$5x^2 - 2x - 4x + 16 = 0$$

Simplifying the equation, we get:

$$5x^2 - 6x + 16 = 0$$

Solving the Quadratic Equation

The equation is a quadratic equation in the form ax^2 + bx + c = 0. We can solve this equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, a = 5, b = -6, and c = 16. Plugging these values into the formula, we get:

$$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(5)(16)}}{2(5)}$$

Simplifying the expression, we get:

$$x = \frac{6 \pm \sqrt{36 - 320}}{10}$$

$$x = \frac{6 \pm \sqrt{-284}}{10}$$

Complex Solutions

The equation has complex solutions. The square root of a negative number is an imaginary number, which means that the solutions will be complex numbers.

Conclusion

In this article, we solved a logarithmic equation that involved the subtraction of two logarithms with the same base and the addition of a logarithm with a different base. We used properties of logarithms to simplify the equation and solve for the variable x. The equation had complex solutions, which means that the solutions will be complex numbers.

Final Answer

The final answer is not a single value, but rather a complex expression:

$$x = \frac{6 \pm \sqrt{-284}}{10}$$

This expression represents the two complex solutions to the equation.

Tips and Tricks

When solving logarithmic equations, it's essential to use properties of logarithms to simplify the equation. This can help to make the equation more manageable and easier to solve. Additionally, when dealing with complex solutions, it's crucial to remember that the square root of a negative number is an imaginary number.

Related Topics

Logarithmic equations can be challenging to solve, but there are many resources available to help. Some related topics include:

• Properties of logarithms
• Solving quadratic equations
• Complex numbers

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Quadratic Equations" by Khan Academy
• [3] "Complex Numbers" by Wolfram MathWorld
  

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Introduction

Logarithmic equations can be challenging to solve, especially when they involve multiple logarithms with different bases. In this article, we will provide a Q&A section to help clarify any questions or concerns you may have about solving 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. Logarithmic equations can be used to solve problems involving growth and decay, as well as to model real-world phenomena.

Q: What are some common properties of logarithms?

A: Some common properties of logarithms include:

• The product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• The quotient rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• The power rule: $\log_b(x^y) = y\log_b(x)$
• The change of base rule: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the following steps:

1. Use properties of logarithms to simplify the equation.
2. Isolate the logarithmic term.
3. Use the definition of a logarithm to rewrite the equation in exponential form.
4. Solve for the variable.

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 inverse operation of a logarithm.

Q: Can logarithmic equations have complex solutions?

A: Yes, logarithmic equations can have complex solutions. This occurs when the argument of the logarithm is negative.

Q: How do I determine if a logarithmic equation has complex solutions?

A: To determine if a logarithmic equation has complex solutions, you can use the following steps:

1. Check if the argument of the logarithm is negative.
2. If the argument is negative, the equation will have complex solutions.

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 properties of logarithms to simplify the equation.
• Not isolating the logarithmic term.
• Not using the definition of a logarithm to rewrite the equation in exponential form.
• Not checking for complex solutions.

Q: How do I check my work when solving logarithmic equations?

A: To check your work when solving logarithmic equations, you can use the following steps:

1. Plug your solution back into the original equation.
2. Simplify the equation to ensure that it is true.
3. Check for complex solutions.

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

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

• Modeling population growth and decay.
• Modeling chemical reactions.
• Modeling financial transactions.
• Modeling physical phenomena such as sound waves and light waves.

Q: Where can I find more information about logarithmic equations?

A: There are many resources available to help you learn more about logarithmic equations, including:

• Online tutorials and videos.
• Textbooks and workbooks.
• Online forums and communities.
• Professional development courses.

Q: How can I practice solving logarithmic equations?

A: There are many ways to practice solving logarithmic equations, including:

• Working through practice problems.
• Using online resources such as Khan Academy and Mathway.
• Joining a study group or online community.
• Taking a course or workshop.

Q: What are some common logarithmic equations that I should know how to solve?

A: Some common logarithmic equations that you should know how to solve include:

• $\log_b(x) = y$
• $\log_b(xy) = z$
• $\log_b(\frac{x}{y}) = z$
• $\log_b(x^y) = z$

Q: How can I use logarithmic equations in my career?

A: Logarithmic equations have many applications in various careers, including:

• Science and engineering.
• Finance and economics.
• Computer science and programming.
• Data analysis and statistics.

Q: What are some common mistakes to avoid when using logarithmic equations in my career?

A: Some common mistakes to avoid when using logarithmic equations in your career include:

• Not using properties of logarithms to simplify the equation.
• Not checking for complex solutions.
• Not using the definition of a logarithm to rewrite the equation in exponential form.
• Not checking your work.

Q: How can I stay up-to-date with the latest developments in logarithmic equations?

A: There are many ways to stay up-to-date with the latest developments in logarithmic equations, including:

• Attending conferences and workshops.
• Reading industry publications and journals.
• Joining online communities and forums.
• Participating in online discussions and webinars.

Q: What are some resources for learning more about logarithmic equations?

A: There are many resources available to help you learn more about logarithmic equations, including:

• Online tutorials and videos.
• Textbooks and workbooks.
• Online forums and communities.
• Professional development courses.

Q: How can I get help with logarithmic equations?

A: There are many ways to get help with logarithmic equations, including:

• Asking a teacher or tutor.
• Joining an online community or forum.
• Using online resources such as Khan Academy and Mathway.
• Attending a workshop or conference.