What Is The Solution To $\log_3(x+12)=\log_3(5x)$?A. $x=-8$ B. $ X = − 3 X=-3 X = − 3 [/tex] C. $x=3$ D. $x=8$

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Introduction

Logarithmic equations can be challenging to solve, especially when they involve different bases. However, with the correct approach, we can simplify and solve these equations. In this article, we will focus on solving the logarithmic equation $\log_3(x+12)=\log_3(5x)$ and find the correct solution among the given options.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's briefly review the properties of logarithmic equations. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The general form of a logarithmic equation is $\log_b(x)=y$, where $b$ is the base of the logarithm, $x$ is the argument, and $y$ is the result.

One of the key properties of logarithmic equations is the one-to-one property, which states that if $\log_b(x)=\log_b(y)$, then $x=y$. This property will be useful in solving the given equation.

Solving the Logarithmic Equation

Now, let's focus on solving the equation $\log_3(x+12)=\log_3(5x)$. Since the bases of the logarithms are the same, we can use the one-to-one property to simplify the equation.

log⁡3(x+12)=log⁡3(5x)\log_3(x+12)=\log_3(5x)

Using the one-to-one property, we can equate the arguments of the logarithms:

x+12=5xx+12=5x

Simplifying the Equation

Now, let's simplify the equation by isolating the variable $x$. We can start by subtracting $x$ from both sides of the equation:

12=4x12=4x

Solving for $x$

Now, let's solve for $x$ by dividing both sides of the equation by $4$:

x=124x=\frac{12}{4}

x=3x=3

Checking the Solution

Before we conclude that $x=3$ is the solution, let's check if it satisfies the original equation. We can substitute $x=3$ into the original equation and verify that it is true:

log⁡3(3+12)=log⁡3(5(3))\log_3(3+12)=\log_3(5(3))

log⁡3(15)=log⁡3(15)\log_3(15)=\log_3(15)

Since the equation is true, we can conclude that $x=3$ is indeed the solution.

Conclusion

In this article, we solved the logarithmic equation $\log_3(x+12)=\log_3(5x)$ and found that the solution is $x=3$. We used the one-to-one property of logarithmic equations to simplify the equation and isolate the variable $x$. We also checked the solution to ensure that it satisfies the original equation.

Final Answer

The final answer is $x=3$.

Discussion

The given options are A. $x=-8$, B. $x=-3$, C. $x=3$, and D. $x=8$. However, based on our solution, we can conclude that the correct answer is C. $x=3$.

Related Topics

  • Logarithmic equations
  • One-to-one property of logarithmic equations
  • Exponentiation
  • Inverse operations

Further Reading

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

References

  • [1] "Logarithmic Equations" by Khan Academy
  • [2] "Logarithmic Equations" by Mathway
  • [3] "Logarithmic Equations" by Wolfram Alpha

Introduction

Logarithmic equations can be challenging to solve, especially for those who are new to the concept. In this article, we will address some of the most frequently asked questions about logarithmic equations and provide clear and concise answers.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The general form of a logarithmic equation is $\log_b(x)=y$, where $b$ is the base of the logarithm, $x$ is the argument, and $y$ is the result.

Q: What is the one-to-one property of logarithmic equations?

A: The one-to-one property of logarithmic equations states that if $\log_b(x)=\log_b(y)$, then $x=y$. This property is useful in solving logarithmic equations, as it allows us to equate the arguments of the logarithms.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the one-to-one property to equate the arguments of the logarithms. Then, you can use algebraic manipulations to isolate the variable. For example, if you have the equation $\log_3(x+12)=\log_3(5x)$, you can equate the arguments to get $x+12=5x$, and then 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 inverse operation of a logarithm. For example, the equation $\log_3(x)=y$ is a logarithmic equation, while the equation $3^y=x$ is an exponential equation.

Q: Can I use logarithmic equations to solve exponential equations?

A: Yes, you can use logarithmic equations to solve exponential equations. By taking the logarithm of both sides of the exponential equation, you can convert it into a logarithmic equation, which can then be solved using the one-to-one property.

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 one-to-one property to equate the arguments of the logarithms
  • Not checking the solution to ensure that it satisfies the original equation
  • Not using the correct base for the logarithm
  • Not simplifying the equation before solving for the variable

Q: How do I check if my solution is correct?

A: To check if your solution is correct, you can substitute the solution back into the original equation and verify that it is true. For example, if you have the equation $\log_3(x+12)=\log_3(5x)$ and you think the solution is $x=3$, you can substitute $x=3$ into the equation and verify that it is true.

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

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

  • Modeling population growth and decay
  • Analyzing financial data and predicting stock prices
  • Solving problems in physics and engineering
  • Modeling the spread of diseases and predicting the number of cases

Conclusion

In this article, we addressed some of the most frequently asked questions about logarithmic equations and provided clear and concise answers. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of logarithmic equations.

Final Answer

The final answer is that logarithmic equations are a powerful tool for solving problems in mathematics and real-world applications.

Discussion

The discussion of logarithmic equations is ongoing, and there are many more topics to explore. Some possible topics for future discussion include:

  • Logarithmic equations with different bases
  • Logarithmic equations with multiple variables
  • Logarithmic equations with complex numbers
  • Logarithmic equations in real-world applications

Related Topics

  • Logarithmic equations
  • One-to-one property of logarithmic equations
  • Exponentiation
  • Inverse operations

Further Reading

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

References

  • [1] "Logarithmic Equations" by Khan Academy
  • [2] "Logarithmic Equations" by Mathway
  • [3] "Logarithmic Equations" by Wolfram Alpha