What Is The True Solution To The Equation Below?$2 \log_3(6x) - \log_3(4x) = 2 \log_3(x+2$\]A. $x = -2$ And $x = 1$ B. $x = -2$ And $x = 2$ C. $x = 1$ And $x = 4$ D. $x = 2$ And

Introduction

In this article, we will delve into the world of logarithmic equations and explore the solution to a given equation. The equation in question is $2 \log_3(6x) - \log_3(4x) = 2 \log_3(x+2)$. We will use various mathematical techniques to simplify the equation and find the true solution.

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 $a^b = c$, then $\log_a(c) = b$. Logarithmic equations can be solved using various techniques, including the use of logarithmic properties and the change of base formula.

Simplifying the Equation

To simplify the equation, we can start by using the property of logarithms that states $\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})$. Applying this property to the given equation, we get:

$$2 \log_3(6x) - \log_3(4x) = 2 \log_3(x+2)$$

$$\log_3(\frac{2(6x)}{4x}) = \log_3(x+2)$$

$$\log_3(3x) = \log_3(x+2)$$

Using the One-to-One Property of Logarithms

Now that we have simplified the equation, we can use the one-to-one property of logarithms to solve for $x$. The one-to-one property states that if $\log_a(b) = \log_a(c)$, then $b = c$. Applying this property to the equation, we get:

$$3x = x+2$$

Solving for $x$

Now that we have a linear equation, we can solve for $x$ by isolating the variable. Subtracting $x$ from both sides of the equation, we get:

$$2x = 2$$

Dividing both sides of the equation by 2, we get:

$$x = 1$$

Checking the Solution

Before we conclude that $x = 1$ is the true solution, we need to check that it satisfies the original equation. Plugging $x = 1$ into the original equation, we get:

$$2 \log_3(6(1)) - \log_3(4(1)) = 2 \log_3(1+2)$$

$$2 \log_3(6) - \log_3(4) = 2 \log_3(3)$$

$$\log_3(36) - \log_3(4) = \log_3(9)$$

$$\log_3(\frac{36}{4}) = \log_3(9)$$

$$\log_3(9) = \log_3(9)$$

Since the equation is true, we can conclude that $x = 1$ is indeed the true solution.

Conclusion

In this article, we explored the solution to the equation $2 \log_3(6x) - \log_3(4x) = 2 \log_3(x+2)$. We used various mathematical techniques, including the use of logarithmic properties and the change of base formula, to simplify the equation and find the true solution. We found that the true solution is $x = 1$.

Final Answer

The final answer is $x = 1$.

Additional Solutions

However, we should also consider the possibility of additional solutions. To do this, we can use the fact that the original equation is an equality, and therefore, the two sides of the equation must be equal for all values of $x$. This means that we can set the two sides of the equation equal to each other and solve for $x$.

Setting the Two Sides Equal

Setting the two sides of the equation equal to each other, we get:

$$2 \log_3(6x) - \log_3(4x) = 2 \log_3(x+2)$$

$$\log_3(36x^2) - \log_3(4x) = \log_3(9(x+2))$$

Using the Quotient Property of Logarithms

Now that we have set the two sides of the equation equal to each other, we can use the quotient property of logarithms to simplify the equation. The quotient property states that $\log_a(\frac{b}{c}) = \log_a(b) - \log_a(c)$. Applying this property to the equation, we get:

$$\log_3(36x^2) - \log_3(4x) = \log_3(9(x+2))$$

$$\log_3(\frac{36x^2}{4x}) = \log_3(9(x+2))$$

$$\log_3(9x) = \log_3(9(x+2))$$

Using the One-to-One Property of Logarithms

Now that we have simplified the equation, we can use the one-to-one property of logarithms to solve for $x$. The one-to-one property states that if $\log_a(b) = \log_a(c)$, then $b = c$. Applying this property to the equation, we get:

$$9x = 9(x+2)$$

Solving for $x$

Now that we have a linear equation, we can solve for $x$ by isolating the variable. Subtracting $9x$ from both sides of the equation, we get:

$$0 = 9x + 18 - 9x$$

Simplifying the equation, we get:

$$0 = 18$$

This is a contradiction, which means that there are no additional solutions.

Conclusion

In this article, we explored the solution to the equation $2 \log_3(6x) - \log_3(4x) = 2 \log_3(x+2)$. We used various mathematical techniques, including the use of logarithmic properties and the change of base formula, to simplify the equation and find the true solution. We found that the true solution is $x = 1$. We also considered the possibility of additional solutions and found that there are no additional solutions.

Final Answer

The final answer is $x = 1$.

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Change of Base Formula" by Math Is Fun
• [3] "One-to-One Property of Logarithms" by Khan Academy
  Q&A: Logarithmic Equations =============================

Introduction

In our previous article, we explored the solution to the equation $2 \log_3(6x) - \log_3(4x) = 2 \log_3(x+2)$. We used various mathematical techniques, including the use of logarithmic properties and the change of base formula, to simplify the equation and find the true solution. In this article, we will answer some 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 $a^b = c$, then $\log_a(c) = b$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use various mathematical techniques, including the use of logarithmic properties and the change of base formula. You can also use the one-to-one property of logarithms to simplify the equation and find the true solution.

Q: What is the one-to-one property of logarithms?

A: The one-to-one property of logarithms states that if $\log_a(b) = \log_a(c)$, then $b = c$. This means that if the logarithms of two numbers are equal, then the numbers themselves must be equal.

Q: How do I use the one-to-one property of logarithms to solve a logarithmic equation?

A: To use the one-to-one property of logarithms to solve a logarithmic equation, you can set the two sides of the equation equal to each other and simplify the equation using the one-to-one property. For example, if you have the equation $\log_3(36x^2) - \log_3(4x) = \log_3(9(x+2))$, you can set the two sides of the equation equal to each other and simplify the equation using the one-to-one property.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to change the base of a logarithm from one base to another. The formula is $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$, where $a$, $b$, and $c$ are positive numbers.

Q: How do I use the change of base formula to solve a logarithmic equation?

A: To use the change of base formula to solve a logarithmic equation, you can change the base of the logarithm from one base to another and simplify the equation using the change of base formula. For example, if you have the equation $\log_3(36x^2) - \log_3(4x) = \log_3(9(x+2))$, you can change the base of the logarithm from base 3 to base 10 and simplify the equation using the change of base formula.

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 of logarithms to simplify the equation
• Not using the change of base formula to change the base of the logarithm
• Not checking the solution to make sure it satisfies the original equation
• Not using the properties of logarithms to simplify the equation

Conclusion

In this article, we answered some frequently asked questions about logarithmic equations. We discussed the one-to-one property of logarithms, the change of base formula, and some common mistakes to avoid when solving logarithmic equations. We hope that this article has been helpful in understanding logarithmic equations and how to solve them.

Final Answer

The final answer is that logarithmic equations can be solved using various mathematical techniques, including the use of logarithmic properties and the change of base formula. By understanding the one-to-one property of logarithms and the change of base formula, you can simplify logarithmic equations and find the true solution.

References

• [1] "Logarithmic Equations" by Math Open Reference
• [2] "Change of Base Formula" by Math Is Fun
• [3] "One-to-One Property of Logarithms" by Khan Academy