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 2 X = \frac{1}{2} X = 2 1 B. X = 3 2 X = \frac{3}{2} X = 2 3 C. X = 1 3 X = \frac{1}{3} X = 3 1 D. X = 1 X = 1 X = 1

Mar 13, 2025 by ADMIN 318 views

**Solving Logarithmic Equations: A Step-by-Step Guide**

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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 base 6, and our goal is to find the value of $x$ that satisfies this equation.

Using Logarithmic Properties

To solve this equation, we can use the properties of logarithms. One of the key properties is the product rule, which states that:

$\log _a (m \cdot n) = \log _a m + \log _a n$

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

$\log _6 (x \cdot 3) = \log _6 (x+1)$

Equating the Arguments

Since the logarithms have the same base, we can equate the arguments of the logarithms:

$x \cdot 3 = x+1$

Solving for x

Now, we can solve for $x$ by isolating it on one side of the equation:

$x \cdot 3 - x = 1$

$x(3-1) = 1$

$x \cdot 2 = 1$

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

Conclusion

In this article, we solved a logarithmic equation using the properties of logarithms and basic algebra. We arrived at the solution $x = \frac{1}{2}$ , which is the correct answer.

Why is this the Correct Answer?

To verify that $x = \frac{1}{2}$ is the correct answer, we can substitute this value back into the original equation:

$\log _6 \left(\frac{1}{2}\right) + \log _6 3 = \log _6 \left(\frac{1}{2}+1\right)$

$\log _6 \left(\frac{1}{2}\right) + \log _6 3 = \log _6 \left(\frac{3}{2}\right)$

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

$\log _6 \left(\frac{1}{2} \cdot 3\right) = \log _6 \left(\frac{3}{2}\right)$

$\log _6 \left(\frac{3}{2}\right) = \log _6 \left(\frac{3}{2}\right)$

This shows that $x = \frac{1}{2}$ is indeed the correct solution to the equation.

What if I Choose a Different Answer?

Let's consider what would happen if we chose a different answer, such as $x = \frac{3}{2}$ .

Substituting this value back into the original equation, we get:

$\log _6 \left(\frac{3}{2}\right) + \log _6 3 = \log _6 \left(\frac{3}{2}+1\right)$

$\log _6 \left(\frac{3}{2}\right) + \log _6 3 = \log _6 \left(\frac{5}{2}\right)$

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

$\log _6 \left(\frac{3}{2} \cdot 3\right) = \log _6 \left(\frac{5}{2}\right)$

$\log _6 \left(\frac{9}{2}\right) = \log _6 \left(\frac{5}{2}\right)$

This shows that $x = \frac{3}{2}$ is not the correct solution to the equation.

Why is this the Incorrect Answer?

The reason why $x = \frac{3}{2}$ is not the correct answer is that it does not satisfy the original equation. When we substitute this value back into the equation, we get a false statement, which means that it is not a valid solution.

Conclusion

In this article, we solved a logarithmic equation using the properties of logarithms and basic algebra. We arrived at the solution $x = \frac{1}{2}$ , which is the correct answer. We also considered what would happen if we chose a different answer, such as $x = \frac{3}{2}$ , and showed that it is not the correct solution.

Final Answer

The final answer is:

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

This is the correct solution to the equation.

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Q: What is the main concept behind solving logarithmic equations?

A: The main concept behind solving logarithmic equations is to use the properties of logarithms, such as the product rule and the power rule, to simplify the equation and isolate the variable.

Q: How do I apply the product rule to solve logarithmic equations?

A: To apply the product rule, you need to rewrite the equation in a way that allows you to use the product rule. For example, if you have an equation like $\log _a (m \cdot n) = \log _a m + \log _a n$ , you can rewrite it as $\log _a (x \cdot 3) = \log _6 (x+1)$ .

Q: What is the power rule for logarithms?

A: The power rule for logarithms states that $\log _a (m^b) = b \cdot \log _a m$ . This rule allows you to simplify logarithmic expressions by bringing the exponent down as a coefficient.

Q: How do I use the power rule to solve logarithmic equations?

A: To use the power rule, you need to rewrite the equation in a way that allows you to use the power rule. For example, if you have an equation like $\log _a (m^b) = c$ , you can rewrite it as $b \cdot \log _a m = c$ .

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

A: A logarithmic equation is an equation that involves a logarithmic expression, whereas a logarithmic expression is a mathematical expression that involves a logarithm. For example, $\log _a x = 2$ is a logarithmic equation, whereas $\log _a x + 2$ is a logarithmic expression.

Q: How do I determine the base of a logarithm?

A: To determine the base of a logarithm, you need to look at the equation and identify the base that is being used. For example, in the equation $\log _6 x = 2$ , the base is 6.

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are related in that they are inverse operations. This means that if you have an equation like $a^b = c$ , you can rewrite it as $\log _a c = b$ .

Q: How do I use logarithms to solve exponential equations?

A: To use logarithms to solve exponential equations, you need to rewrite the equation in a way that allows you to use logarithms. For example, if you have an equation like $a^b = c$ , you can rewrite it as $\log _a c = b$ .

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 isolating the variable correctly
Not checking the solution to make sure it satisfies the original equation

Q: How do I check my solution to a logarithmic equation?

A: To check your solution to a logarithmic equation, you need to substitute the solution back into the original equation and make sure it is true. For example, if you have an equation like $\log _a x = 2$ and you find that $x = a^2$ , you need to substitute $x = a^2$ back into the original equation and make sure it is true.

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 do I use logarithmic equations to model real-world phenomena?

A: To use logarithmic equations to model real-world phenomena, you need to identify the key variables and relationships involved in the phenomenon. For example, if you are modeling population growth, you need to identify the initial population, the growth rate, and the time period involved.

Q: What are some common types of logarithmic equations?

A: Some common types of logarithmic equations include:

Linear logarithmic equations: These are equations of the form $\log _a x = b$ .
Quadratic logarithmic equations: These are equations of the form $\log _a x^2 = b$ .
Exponential logarithmic equations: These are equations of the form $\log _a a^b = c$ .

Q: How do I solve logarithmic equations with different bases?

A: To solve logarithmic equations with different bases, you need to use the change of base formula, which states that $\log _a x = \frac{\log _b x}{\log _b a}$ .

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

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

Not using the correct change of base formula
Not simplifying the equation correctly
Not checking the solution to make sure it satisfies the original equation

Q: How do I use logarithmic equations to solve problems involving rates and ratios?

A: To use logarithmic equations to solve problems involving rates and ratios, you need to identify the key variables and relationships involved in the problem. For example, if you are solving a problem involving the rate of change of a quantity, you need to identify the initial quantity, the rate of change, and the time period involved.

Q: What are some common types of problems that involve logarithmic equations?

A: Some common types of problems that involve logarithmic equations include:

Problems involving population growth and decay
Problems involving interest rates and investment returns
Problems involving rates and ratios

Q: How do I use logarithmic equations to solve problems involving optimization?

A: To use logarithmic equations to solve problems involving optimization, you need to identify the key variables and relationships involved in the problem. For example, if you are solving a problem involving the optimization of a function, you need to identify the function, the variables involved, and the constraints.

Q: What are some common types of problems that involve logarithmic equations and optimization?

A: Some common types of problems that involve logarithmic equations and optimization include:

Problems involving the optimization of a function
Problems involving the minimization or maximization of a quantity
Problems involving the optimization of a system

Q: How do I use logarithmic equations to solve problems involving systems of equations?

A: To use logarithmic equations to solve problems involving systems of equations, you need to identify the key variables and relationships involved in the problem. For example, if you are solving a problem involving a system of linear equations, you need to identify the equations, the variables involved, and the constraints.

Q: What are some common types of problems that involve logarithmic equations and systems of equations?

A: Some common types of problems that involve logarithmic equations and systems of equations include:

Problems involving a system of linear equations
Problems involving a system of nonlinear equations
Problems involving a system of differential equations