Solve The Equation: Log ⁡ 3 X + Log ⁡ 3 ( 2 X + 3 ) = 2 \log_3 X + \log_3(2x + 3) = 2 Lo G 3 ​ X + Lo G 3 ​ ( 2 X + 3 ) = 2

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 $\log_3 x + \log_3(2x + 3) = 2$. This equation involves logarithms with the same base, and we will use properties of logarithms to simplify and solve it.

Understanding Logarithmic Properties

Before we dive into solving the equation, let's review some essential properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In other words, if $y = \log_b x$, then $b^y = x$. This property is the foundation of logarithmic equations.

The Product Property

One of the most useful properties of logarithms is the product property, which states that $\log_b (xy) = \log_b x + \log_b y$. This property allows us to combine two logarithmic expressions into a single logarithmic expression.

The Quotient Property

Another important property of logarithms is the quotient property, which states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This property enables us to simplify logarithmic expressions involving division.

The Power Property

The power property of logarithms states that $\log_b x^n = n \log_b x$. This property allows us to simplify logarithmic expressions involving exponents.

Solving the Equation

Now that we have reviewed the essential properties of logarithms, let's focus on solving the equation $\log_3 x + \log_3(2x + 3) = 2$. To simplify this equation, we will use the product property of logarithms.

Step 1: Combine the Logarithmic Expressions

Using the product property, we can combine the two logarithmic expressions on the left-hand side of the equation:

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

Step 2: Simplify the Equation

Now that we have combined the logarithmic expressions, we can simplify the equation:

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

Step 3: Eliminate the Logarithm

To eliminate the logarithm, we will raise both sides of the equation to the power of 3:

$$3^{\log_3 (x(2x + 3))} = 3^2$$

Step 4: Simplify the Equation

Using the power property of logarithms, we can simplify the equation:

$$x(2x + 3) = 9$$

Step 5: Expand and Simplify

Expanding and simplifying the equation, we get:

$$2x^2 + 3x - 9 = 0$$

Step 6: Solve the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

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

In this case, $a 2$, $b = 3$, and $c = -9$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-9)}}{2(2)}$$

Step 7: Simplify the Equation

Simplifying the equation, we get:

$$x = \frac{-3 \pm \sqrt{81}}{4}$$

Step 8: Solve for x

Solving for x, we get two possible solutions:

$$x = \frac{-3 + 9}{4} = 1.5$$

$$x = \frac{-3 - 9}{4} = -3$$

Step 9: Check the Solutions

To check the solutions, we need to plug them back into the original equation. If the solutions satisfy the equation, then they are valid.

Plugging $x = 1.5$ into the original equation, we get:

$$\log_3 (1.5) + \log_3(2(1.5) + 3) = \log_3 (1.5) + \log_3(6)$$

Using the product property, we can combine the logarithmic expressions:

$$\log_3 (1.5) + \log_3(6) = \log_3 (1.5(6))$$

Simplifying the equation, we get:

$$\log_3 (9) = 2$$

This is true, so $x = 1.5$ is a valid solution.

Plugging $x = -3$ into the original equation, we get:

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

Using the product property, we can combine the logarithmic expressions:

$$\log_3 (-3) + \log_3(-3) = \log_3 (-3(-3))$$

Simplifying the equation, we get:

$$\log_3 (9) = 2$$

This is true, so $x = -3$ is also a valid solution.

Conclusion

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Introduction

In our previous article, we solved the equation $\log_3 x + \log_3(2x + 3) = 2$ using properties of logarithms. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to solve logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves logarithms. Logarithms are the inverse of exponents, and they are used to solve equations that involve exponential expressions.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• The product property: $\log_b (xy) = \log_b x + \log_b y$
• The quotient property: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
• The power property: $\log_b x^n = n \log_b x$

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the following steps:

1. Combine the logarithmic expressions using the product property.
2. Eliminate the logarithm by raising both sides to the power of the base.
3. Simplify the equation and solve for the variable.

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

A: A logarithmic equation involves logarithms, while an exponential equation involves exponents. Logarithmic equations are used to solve equations that involve exponential expressions, while exponential equations are used to solve equations that involve logarithmic expressions.

Q: Can I use logarithmic equations to solve equations with different bases?

A: Yes, you can use logarithmic equations to solve equations with different bases. However, you will need to use the change of base formula to convert the equation to a common base.

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

A: To check your solutions to a logarithmic equation, you can plug the solutions back into the original equation and verify that they satisfy the equation.

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

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

• Forgetting to combine the logarithmic expressions using the product property.
• Forgetting to eliminate the logarithm by raising both sides to the power of the base.
• Making errors when simplifying the equation.

Q: Can I use logarithmic equations to solve equations with multiple variables?

A: Yes, you can use logarithmic equations to solve equations with multiple variables. However, you will need to use the properties of logarithms to simplify the equation and solve for the variables.

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

A: Logarithmic equations can be applied to a wide range of real-world problems, including:

• Finance: Logarithmic equations can be used to calculate interest rates and investment returns.
• Science: Logarithmic equations can be used to model population and decay.
• Engineering: Logarithmic equations can be used to design and optimize systems.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and techniques used to solve logarithmic equations. We have covered topics such as the properties of logarithms, how to solve logarithmic equations, and how to check solutions. We have also discussed common mistakes to avoid and how to apply logarithmic equations to real-world problems.