Select The Correct Answer.What Is The Solution To This Equation? Log ⁡ 6 X + Log ⁡ 6 3 = Log ⁡ 6 ( X + 1 \log_6 X + \log_6 3 = \log_6(x+1 Lo G 6 ​ X + Lo G 6 ​ 3 = Lo G 6 ​ ( X + 1 ]A. X = 1 3 X = \frac{1}{3} X = 3 1 ​ B. X = 1 X = 1 X = 1 C. X = 3 2 X = \frac{3}{2} X = 2 3 ​ D. X = 1 2 X = \frac{1}{2} X = 2 1 ​

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific logarithmic equation and provide a step-by-step guide on how to arrive at the correct solution.

The Equation

The given equation is:

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

This equation involves logarithms with the same base, which makes it a good candidate for using logarithmic properties to simplify and solve.

Using Logarithmic Properties

One of the key properties of logarithms is the product rule, which states that:

$\log_b (m \cdot n) = \log_b m + \log_b n$

Using this property, we can rewrite the given equation as:

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

Simplifying the Equation

Now that we have applied the product rule, we can simplify the equation further by combining the terms inside the logarithm:

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

Equating the Arguments

Since the logarithms have the same base, we can equate the arguments (the expressions inside the logarithms):

$3x = x+1$

Solving for x

Now that we have a linear equation, we can solve for x by isolating the variable:

$3x - x = 1$

$2x = 1$

$x = \frac{1}{2}$

Conclusion

In this article, we have solved a logarithmic equation using logarithmic properties and algebraic manipulations. By applying the product rule and simplifying the equation, we arrived at the correct solution, which is:

$x = \frac{1}{2}$

This solution is consistent with option D, which is the correct answer.

Why is this solution correct?

The solution is correct because it satisfies the original equation. By substituting x = 1/2 into the original equation, we get:

$\log_6 (1/2) + \log_6 3 = \log_6(1/2 + 1)$

$\log_6 (1/2) + \log_6 3 = \log_6(3/2)$

Using the product rule, we can rewrite the left-hand side as:

$\log_6 (1/2 \cdot 3) = \log_6(3/2)$

$\log_6 (3/2) = \log_6(3/2)$

This shows that the solution x = 1/2 satisfies the original equation, making it the correct answer.

Final Thoughts

Q: What is the product rule for logarithms?

A: The product rule for logarithms states that:

$\log_b (m \cdot n) = \log_b m + \log_b n$

This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do I apply the product rule to a logarithmic equation?

A: To apply the product rule, simply rewrite the equation using the product rule formula:

$\log_b (m \cdot n) = \log_b m + \log_b n$

For example, if we have the equation:

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

We can rewrite it as:

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

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithm, whereas an exponential equation is an equation that involves an exponential function. For example:

$\log_6 x = 2$ is a logarithmic equation

$6^x = 36$ is an exponential equation

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, follow these steps:

1. Apply the product rule to simplify the equation
2. Equate the arguments (the expressions inside the logarithms)
3. Solve for the variable using algebraic manipulations

For example, if we have the equation:

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

We can solve it by following these steps:

1. Apply the product rule: $\log_6 (x \cdot 3) = \log_6 x + \log_6 3$
2. Equate the arguments: $x \cdot 3 = x+1$
3. Solve for x: $2x = 1$, $x = \frac{1}{2}$

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is used as the exponent in the exponential function. For example, in the equation:

$\log_6 x = 2$

The base is 6, because the exponential function is:

$6^x$

Q: Can I use a calculator to solve a logarithmic equation?

A: Yes, you can use a calculator to solve a logarithmic equation. However, it's always a good idea to check your work by plugging the solution back into the original equation to make sure it's true.

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

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

• Not applying the product rule correctly
• Not equating the arguments correctly
• Not solving for the variable correctly
• Not checking the solution by plugging it back into the original equation

By avoiding these common mistakes, you can ensure that you're solving logarithmic equations correctly and accurately.