Solve The Following Logarithmic Equation.$\log _6(2x) = 3$A. $x = 12$ B. $x = 21$ C. $x = 364$ D. $x = 108$

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Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of logarithms. In this article, we will focus on solving a specific logarithmic equation, log⁡6(2x)=3\log _6(2x) = 3, and explore the different methods and techniques used to find the solution.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function, and it is used to solve equations that involve exponential expressions. In this case, we are given the equation log⁡6(2x)=3\log _6(2x) = 3, where xx is the variable we need to solve for.

The Properties of Logarithms

Before we can solve the equation, we need to understand the properties of logarithms. The logarithmic function has several properties that we can use to simplify the equation. These properties include:

  • The product rule: log⁡a(xy)=log⁡a(x)+log⁡a(y)\log _a(xy) = \log _a(x) + \log _a(y)
  • The quotient rule: log⁡a(xy)=log⁡a(x)−log⁡a(y)\log _a(\frac{x}{y}) = \log _a(x) - \log _a(y)
  • The power rule: log⁡a(xy)=ylog⁡a(x)\log _a(x^y) = y\log _a(x)

Solving the Equation

Now that we have a good understanding of the properties of logarithms, we can use them to solve the equation. We start by rewriting the equation in exponential form:

63=2x6^3 = 2x

We can simplify this equation by evaluating the exponential expression:

216=2x216 = 2x

Next, we can divide both sides of the equation by 2 to solve for xx:

x=2162x = \frac{216}{2}

x=108x = 108

Conclusion

In this article, we have solved the logarithmic equation log⁡6(2x)=3\log _6(2x) = 3 using the properties of logarithms. We have shown that the solution to the equation is x=108x = 108. This is a fundamental concept in mathematics, and it is essential to understand how to solve logarithmic equations in order to solve more complex problems.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes to avoid. These include:

  • Not using the correct properties of logarithms: Make sure to use the product rule, quotient rule, and power rule correctly.
  • Not rewriting the equation in exponential form: This is a crucial step in solving logarithmic equations.
  • Not simplifying the equation: Make sure to simplify the equation as much as possible before solving for the variable.

Real-World Applications

Logarithmic equations have many real-world applications. Some examples include:

  • 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 and optimize systems.

Practice Problems

Here are some practice problems to help you reinforce your understanding of logarithmic equations:

  • Problem 1: Solve the equation log⁡5(3x)=2\log _5(3x) = 2.
  • Problem 2: Solve the equation log⁡2(x3)=5\log _2(x^3) = 5.
  • Problem 3: Solve the equation log⁡3(x2)=4\log _3(\frac{x}{2}) = 4.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about logarithmic equations. Whether you are a student, teacher, or simply someone looking to learn more about logarithmic equations, this article is for you.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function, and it is used to solve equations that involve exponential expressions.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use the properties of logarithms. The most common properties of logarithms are the product rule, quotient rule, and power rule. You can use these properties to rewrite the equation in exponential form and then solve for the variable.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that log⁡a(xy)=log⁡a(x)+log⁡a(y)\log _a(xy) = \log _a(x) + \log _a(y). This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that log⁡a(xy)=log⁡a(x)−log⁡a(y)\log _a(\frac{x}{y}) = \log _a(x) - \log _a(y). This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that log⁡a(xy)=ylog⁡a(x)\log _a(x^y) = y\log _a(x). This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I rewrite a logarithmic equation in exponential form?

A: To rewrite a logarithmic equation in exponential form, you need to use the definition of a logarithm. The definition of a logarithm states that log⁡a(x)=y\log _a(x) = y is equivalent to ay=xa^y = x. You can use this definition to rewrite the logarithmic equation in exponential form.

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

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

  • Not using the correct properties of logarithms
  • Not rewriting the equation in exponential form
  • Not simplifying the equation
  • Not checking the domain of the logarithmic function

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

A: Logarithmic equations have many 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 and optimize systems.

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 and calculators
  • Joining a study group or seeking help from a tutor
  • Taking online courses or watching video tutorials

Conclusion

In conclusion, logarithmic equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of logarithms and the ability to apply them correctly. By following the steps outlined in this article and practicing regularly, you can become proficient in solving logarithmic equations and apply them to real-world problems.