Solve The Equation:${ \log _6 3x + \log _6(x-1) = 3 }$

Mar 13, 2025 by ADMIN 56 views

Solving the Logarithmic Equation: 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 the equation $\log _6 3x + \log _6(x-1) = 3$ . This equation involves logarithms with the same base, and we will use properties of logarithms to simplify and solve it.

Understanding Logarithms

Before we dive into solving the equation, let's quickly review the basics of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$ , then $\log_a c = b$ . The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number.

Properties of Logarithms

There are several properties of logarithms that we will use to simplify the equation. These properties include:

Product Property: $\log_a (xy) = \log_a x + \log_a y$
Quotient Property: $\log_a \frac{x}{y} = \log_a x - \log_a y$
Power Property: $\log_a x^b = b \log_a x$

Simplifying the Equation

Now that we have reviewed the basics of logarithms and their properties, let's simplify the equation. We can start by combining the two logarithmic terms on the left-hand side of the equation using the product property:

\log _6 3x + \log _6(x-1) = \log _6 (3x(x-1))

This simplifies the equation to:

\log _6 (3x(x-1)) = 3

Using the Definition of Logarithm

We can now use the definition of logarithm to rewrite the equation in exponential form:

6^3 = 3x(x-1)

This simplifies to:

216 = 3x(x-1)

Solving the Quadratic Equation

We can now solve the quadratic equation by expanding the right-hand side and rearranging the terms:

216 = 3x^2 - 3x

Rearranging the terms, we get:

3x^2 - 3x - 216 = 0

Factoring the Quadratic Equation

We can now factor the quadratic equation:

(3x + 24)(x - 9) = 0

Finding the Solutions

We can now find the solutions by setting each factor equal to zero:

3x + 24 = 0 \Rightarrow x = -8

x - 9 = 0 \Rightarrow x = 9

Checking the Solutions

We need to check if both solutions satisfy the original equation. We can do this by plugging each solution back into the original equation:

\log _6 3(-8) + \log _6(-8-1) = 3

This simplifies to:

\log _6 (-24) + \log _6(-9) = 3

This is not true, so $x = -8$ is not a valid solution.

\log _6 3(9) + \log _6(9-1) = 3

This simplifies to:

\log _6 (27) + \log _6(8) = 3

This is true, so $x = 9$ is a valid solution.

Conclusion

In this article, we have solved the logarithmic equation $\log _6 3x + \log _6(x-1) = 3$ . We used properties of logarithms to simplify the equation and then solved the resulting quadratic equation. We found that the only valid solution is $x = 9$ .

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Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves logarithms. It is an equation that can be written in the form $\log_a x = b$ , where $a$ is the base of the logarithm, $x$ is the argument of the logarithm, and $b$ is the result of the logarithm.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use the properties of logarithms to simplify the equation and then solve for the variable. You can use the product property, quotient property, and power property of logarithms to simplify the equation.

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_a (xy) = \log_a x + \log_a y$ . This means that the logarithm of a product is equal to the sum of the logarithms of the factors.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_a \frac{x}{y} = \log_a x - \log_a y$ . This means that the logarithm of a quotient is equal to the difference of the logarithms of the dividend and the divisor.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log_a x^b = b \log_a x$ . This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I check if a solution is valid?

A: To check if a solution is valid, you need to plug the solution back into the original equation and check if it is true. If the solution satisfies the original equation, then it is a valid solution.

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

A: A logarithmic equation is an equation that involves logarithms, while an exponential equation is an equation that involves exponents. For example, the equation $\log_a x = b$ is a logarithmic equation, while the equation $a^x = b$ is an exponential equation.

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

A: Yes, you can use a calculator to solve a logarithmic equation. However, you need to make sure that the calculator is set to the correct base and that you are using the correct operation.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to use a graphing calculator or a graphing software. You can also use a table of values to create a graph.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers greater than zero. This means that the argument of the logarithm must be greater than zero.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is all real numbers. This means that the result of the logarithm can be any real number.

Q: Can I use logarithmic equations in real-world applications?

A: Yes, logarithmic equations can be used in real-world applications such as finance, science, and engineering. For example, logarithmic equations can be used to model population growth, chemical reactions, and financial transactions.

Q: How do I apply logarithmic equations to real-world problems?

A: To apply logarithmic equations to real-world problems, you need to identify the variables and the relationships between them. You can then use the properties of logarithms to simplify the equation and solve for the variable.

Q: What are some common applications of logarithmic equations?

A: Some common applications of logarithmic equations include:

Modeling population growth
Chemical reactions
Financial transactions
Sound levels
Light intensity

Q: How do I use logarithmic equations to model population growth?

A: To use logarithmic equations to model population growth, you need to identify the variables and the relationships between them. You can then use the properties of logarithms to simplify the equation and solve for the variable.

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

A: A linear equation is an equation that involves a linear function, while a logarithmic equation is an equation that involves a logarithmic function. For example, the equation $y = 2x + 3$ is a linear equation, while the equation $\log_a x = b$ is a logarithmic equation.

Q: Can I use logarithmic equations to model exponential growth?

A: Yes, you can use logarithmic equations to model exponential growth. However, you need to use the properties of logarithms to simplify the equation and solve for the variable.

Q: How do I use logarithmic equations to model exponential decay?

A: To use logarithmic equations to model exponential decay, you need to identify the variables and the relationships between them. You can then use the properties of logarithms to simplify the equation and solve for the variable.

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

A: A logarithmic equation is an equation that involves logarithms, while a trigonometric equation is an equation that involves trigonometric functions. For example, the equation $\log_a x = b$ is a logarithmic equation, while the equation $\sin x = \frac{1}{2}$ is a trigonometric equation.

Q: Can I use logarithmic equations to model periodic phenomena?

A: Yes, you can use logarithmic equations to model periodic phenomena. However, you need to use the properties of logarithms to simplify the equation and solve for the variable.

Q: How do I use logarithmic equations to model periodic phenomena?

A: To use logarithmic equations to model periodic phenomena, you need to identify the variables and the relationships between them. You can then use the properties of logarithms to simplify the equation and solve for the variable.

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

A: A logarithmic equation is an equation that involves logarithms, while a polynomial equation is an equation that involves polynomials. For example, the equation $\log_a x = b$ is a logarithmic equation, while the equation $x^2 + 3x - 4 = 0$ is a polynomial equation.

Q: Can I use logarithmic equations to model polynomial growth?

A: Yes, you can use logarithmic equations to model polynomial growth. However, you need to use the properties of logarithms to simplify the equation and solve for the variable.

Q: How do I use logarithmic equations to model polynomial growth?

A: To use logarithmic equations to model polynomial growth, you need to identify the variables and the relationships between them. You can then use the properties of logarithms to simplify the equation and solve for the variable.

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

A: A logarithmic equation is an equation that involves logarithms, while a rational equation is an equation that involves rational expressions. For example, the equation $\log_a x = b$ is a logarithmic equation, while the equation $\frac{x+1}{x-1} = 2$ is a rational equation.

Q: Can I use logarithmic equations to model rational growth?

A: Yes, you can use logarithmic equations to model rational growth. However, you need to use the properties of logarithms to simplify the equation and solve for the variable.

Q: How do I use logarithmic equations to model rational growth?

A: To use logarithmic equations to model rational growth, you need to identify the variables and the relationships between them. You can then use the properties of logarithms to simplify the equation and solve for the variable.

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

A: A logarithmic equation is an equation that involves logarithms, while a radical equation is an equation that involves radicals. For example, the equation $\log_a x = b$ is a logarithmic equation, while the equation $\sqrt{x} = 2$ is a radical equation.

Q: Can I use logarithmic equations to model radical growth?

A: Yes, you can use logarithmic equations to model radical growth. However, you need to use the properties of logarithms to simplify the equation and solve for the variable.

Q: How do I use logarithmic equations to model radical growth?

A: To use logarithmic equations to model radical growth, you need to identify the variables and the relationships between them. You can then use the properties of logarithms to simplify the equation and solve for the variable.

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

Q: Can I use logarithmic equations to model periodic phenomena?

A: Yes, you can use logarithmic equations to model periodic phenomena. However, you need to use the properties of logarithms to simplify the equation and solve for the variable.