Solve The Equation.${ \log_6 (3x) + \log_6 (x-1) = 3 }${ X =\$}

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Introduction

In this article, we will delve into solving a logarithmic equation involving base 6 logarithms. The equation is $\log_6 (3x) + \log_6 (x-1) = 3$. Our goal is to find the value of $x$ that satisfies this equation. We will use various properties of logarithms to simplify the equation and ultimately solve for $x$.

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 logarithmic function. The general form of a logarithmic equation is $\log_b (x) = y$, where $b$ is the base of the logarithm and $x$ is the argument of the logarithm. The logarithmic function is the inverse of the exponential function, and it is used to solve equations involving exponential functions.

Simplifying the Equation

To simplify the equation, we can use the property of logarithms that states $\log_b (m) + \log_b (n) = \log_b (mn)$. Applying this property to the given equation, we get:

$$\log_6 (3x) + \log_6 (x-1) = \log_6 (3x(x-1))$$

Using the Definition of Logarithms

We can rewrite the equation using the definition of logarithms. The definition of a logarithm states that if $y = \log_b (x)$, then $b^y = x$. Applying this definition to the equation, we get:

$$6^3 = 3x(x-1)$$

Expanding and Simplifying

We can expand and simplify the equation by multiplying the terms on the right-hand side:

$$216 = 3x^2 - 3x$$

Rearranging the Equation

We can rearrange the equation to get a quadratic equation in terms of $x$:

$$3x^2 - 3x - 216 = 0$$

Solving the Quadratic Equation

We can solve the quadratic equation using the quadratic formula:

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

In this case, $a = 3$, $b = -3$, and $c = -216$. Plugging these values into the formula, we get:

$$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(3)(-216)}}{2(3)}$$

Simplifying the Quadratic Formula

We can simplify the quadratic formula by evaluating the expressions inside the square root:

$$x = \frac{3 \pm \sqrt{9 + 2592}}{6}$$

$$x = \frac{3 \pm \sqrt{2601}}{6}$$

Evaluating the Square Root

We can evaluate the square root by finding the square root of 2601:

$$x = \frac{3 \pm 51}{6}$$

Finding the Solutions

We can find the solutions to the equation by evaluating the two possible values of $x$:

$$x = \frac{3 + 51}{6} = \frac{54}{6} = 9$$

$$x = \frac{3 - 51}{6} = \frac{-48}{6} = -8$$

Checking the Solutions

We can check the solutions by plugging them back into the original equation:

$$\log_6 (3(9)) + \log_6 (9-1) = 3$$

$$\log_6 (27) + \log_6 (8) = 3$$

$$3 + 1 = 3$$

This solution checks out.

$$\log_6 (3(-8)) + \log_6 (-8-1) = 3$$

$$\log_6 (-24) + \log_6 (-9) = 3$$

This solution does not check out, since the logarithm of a negative number is undefined.

Conclusion

In this article, we solved the logarithmic equation $\log_6 (3x) + \log_6 (x-1) = 3$. We used various properties of logarithms to simplify the equation and ultimately solve for $x$. We found that the solution to the equation is $x = 9$. We also checked the solution by plugging it back into the original equation and found that it checks out.

Final Answer

The final answer is $\boxed{9}$.

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Introduction

In our previous article, we solved the logarithmic equation $\log_6 (3x) + \log_6 (x-1) = 3$. We used various properties of logarithms to simplify the equation and ultimately solve for $x$. In this article, we will answer some frequently asked questions about solving this equation.

Q: What is the base of the logarithm in this equation?

A: The base of the logarithm in this equation is 6.

Q: What is the property of logarithms used to simplify the equation?

A: The property of logarithms used to simplify the equation is $\log_b (m) + \log_b (n) = \log_b (mn)$.

Q: How do we rewrite the equation using the definition of logarithms?

A: We can rewrite the equation using the definition of logarithms by applying the property $b^y = x$ to the equation.

Q: What is the quadratic equation obtained after simplifying the equation?

A: The quadratic equation obtained after simplifying the equation is $3x^2 - 3x - 216 = 0$.

Q: How do we solve the quadratic equation?

A: We can solve the quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What are the solutions to the equation?

A: The solutions to the equation are $x = 9$ and $x = -8$.

Q: Which solution checks out?

A: The solution $x = 9$ checks out, while the solution $x = -8$ does not.

Q: Why does the solution $x = -8$ not check out?

A: The solution $x = -8$ does not check out because the logarithm of a negative number is undefined.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = 9$.

Q: Can you provide a step-by-step solution to the equation?

A: Yes, we can provide a step-by-step solution to the equation. Here it is:

1. Simplify the equation using the property of logarithms: $\log_6 (3x) + \log_6 (x-1) = \log_6 (3x(x-1))$
2. Rewrite the equation using the definition of logarithms: $6^3 = 3x(x-1)$
3. Expand and simplify the equation: $216 = 3x^2 - 3x$
4. Rearrange the equation to get a quadratic equation: $3x^2 - 3x - 216 = 0$
5. Solve the quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
6. Evaluate the solutions: $x = \frac{3 \pm \sqrt{2601}}{6}$
7. Simplify the solutions: $x = \frac{3 \pm 51}{6}$
8. Find the solutions: $x = 9$ and $x = -8$
9. Check the solutions: $x = 9$ checks out, while $x = -8$ does not.

Q: Can you provide a video or animation to help understand the solution?

A: Yes, we can provide a video or animation to help understand the solution. However, we do not have the capability to create videos or animations at this time.

Q: Can you provide a practice problem to help reinforce the solution?

A: Yes, we can provide a practice problem to help reinforce the solution. Here it is:

Solve the equation $\log_4 (2x) + \log_4 (x-2) = 2$.

Conclusion

In this article, we answered some frequently asked questions about solving the equation $\log_6 (3x) + \log_6 (x-1) = 3$. We provided step-by-step solutions, explanations, and examples to help reinforce the solution. We also provided a practice problem to help reinforce the solution.