Find The Largest Value Of $x$ That Satisfies: Log ⁡ 3 ( X 2 ) − Log ⁡ 3 ( X + 2 ) = 1 \log_3\left(x^2\right) - \log_3(x+2) = 1 Lo G 3 ​ ( X 2 ) − Lo G 3 ​ ( X + 2 ) = 1

Introduction

Logarithmic equations can be challenging to solve, especially when they involve multiple logarithmic terms. In this article, we will focus on solving the equation $\log_3\left(x^2\right) - \log_3(x+2) = 1$. 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 important properties of logarithms. The logarithm of a number $x$ with base $b$ is denoted as $\log_b(x)$. The logarithm of a product is equal to the sum of the logarithms, and the logarithm of a quotient is equal to the difference of the logarithms. Mathematically, this can be expressed as:

$\log_b(xy) = \log_b(x) + \log_b(y)$

$\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

Simplifying the Equation

Now that we have reviewed the properties of logarithms, let's simplify the given equation. We can start by using the property of logarithms that states $\log_b(x^a) = a\log_b(x)$. Applying this property to the equation, we get:

$\log_3\left(x^2\right) - \log_3(x+2) = 1$

$2\log_3(x) - \log_3(x+2) = 1$

Using the Quotient Property

Next, we can use the quotient property of logarithms to simplify the equation further. The quotient property states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$. Applying this property to the equation, we get:

$2\log_3(x) - \log_3(x+2) = 1$

$\log_3\left(\frac{x^2}{x+2}\right) = 1$

Exponentiating Both Sides

Now that we have simplified the equation, we can exponentiate both sides to get rid of the logarithm. Since the base of the logarithm is 3, we can raise 3 to the power of both sides of the equation. This gives us:

$3^{\log_3\left(\frac{x^2}{x+2}\right)} = 3^1$

$\frac{x^2}{x+2} = 3$

Cross-Multiplying

To solve for $x$, we can cross-multiply the equation. This gives us:

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

$x^2 = 3x + 6$

Rearranging the Equation

Now that we have cross-multiplied the equation, we can rearrange it to get a quadratic equation in standard form. Subtracting $3x + 6$ from both sides of the equation, we get:

$x^2 - 3x - 6 = 0$

Factoring the Quadratic Equation

The quadratic equation $x^2 - 3x - 6 = 0$ can be factored as:

$(x - 6)(x + 1) = 0$

Solving for x

To solve for $x$, we can set each factor equal to zero and solve for $x$. This gives us:

$x - 6 = 0 \Rightarrow x = 6$

$x + 1 = 0 \Rightarrow x = -1$

Checking the Solutions

Before we conclude that $x = 6$ and $x = -1$ are the solutions to the equation, we need to check if they satisfy the original equation. Plugging $x = 6$ into the original equation, we get:

$\log_3\left(6^2\right) - \log_3(6+2) = 1$

$\log_3(36) - \log_3(8) = 1$

$2\log_3(6) - \log_3(8) = 1$

$\log_3\left(\frac{36}{8}\right) = 1$

$\log_3(4.5) = 1$

Since $\log_3(4.5) \neq 1$, $x = 6$ is not a solution to the equation.

Plugging $x = -1$ into the original equation, we get:

$\log_3\left((-1)^2\right) - \log_3(-1+2) = 1$

$\log_3(1) - \log_3(1) = 1$

$0 = 1$

Since $0 \neq 1$, $x = -1$ is not a solution to the equation.

Conclusion

In this article, we solved the logarithmic equation $\log_3\left(x^2\right) - \log_3(x+2) = 1$. We used properties of logarithms to simplify the equation and then solved for $x$. We found that the equation has no real solutions, but we can find the largest value of $x$ that satisfies the equation by analyzing the graph of the function $f(x) = \log_3\left(\frac{x^2}{x+2}\right)$. The largest value of $x$ that satisfies the equation is $x = 6$, but this is not a solution to the original equation. Therefore, the largest value of $x$ that satisfies the equation is actually $x = \boxed{6}$.

Graphing the Function

To find the largest value of $x$ that satisfies the equation, we can graph the function $f(x) = \log_3\left(\frac{x^2}{x+2}\right)$. The graph of the function is a logarithmic curve that approaches negative infinity as $x$ approaches negative infinity and approaches positive infinity as $x$ approaches positive infinity. The graph of the function has a vertical asymptote at $x = -2$ and a horizontal asymptote at $y = -\infty$.

Finding the Largest Value of x

To find the largest value of $x$ that satisfies the equation, we can look at the graph of the function and find the point where the graph is closest to the horizontal line $y = 1$. This point is approximately $x = 6$, but this is not a solution to the original equation. Therefore, the largest value of $x$ that satisfies the equation is actually $x = \boxed{6}$.

Conclusion

In this article, we solved the logarithmic equation $\log_3\left(x^2\right) - \log_3(x+2) = 1$. We used properties of logarithms to simplify the equation and then solved for $x$. We found that the equation has no real solutions, but we can find the largest value of $x$ that satisfies the equation by analyzing the graph of the function $f(x) = \log_3\left(\frac{x^2}{x+2}\right)$. The largest value of $x$ that satisfies the equation is $x = 6$, but this is not a solution to the original equation. Therefore, the largest value of $x$ that satisfies the equation is actually $x = \boxed{6}$.

Introduction

In our previous article, we solved the logarithmic equation $\log_3\left(x^2\right) - \log_3(x+2) = 1$. We used properties of logarithms to simplify the equation and then solved for $x$. In this article, we will answer some common questions that readers may have about solving logarithmic equations.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponential function. For example, the equation $\log_3(x) = 2$ is a logarithmic equation, while the equation $3^x = 9$ is an exponential equation.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the properties of logarithms. For example, if you have the equation $\log_3(x^2) - \log_3(x+2) = 1$, you can use the property $\log_b(x^a) = a\log_b(x)$ to simplify the equation.

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

A: A logarithmic function is a function that involves a logarithm, while an exponential function is a function that involves an exponential expression. For example, the function $f(x) = \log_3(x)$ is a logarithmic function, while the function $f(x) = 3^x$ is an exponential function.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to simplify the equation and then solve for the variable. For example, if you have the equation $\log_3(x^2) - \log_3(x+2) = 1$, you can use the property $\log_b(x^a) = a\log_b(x)$ to simplify the equation and then solve for $x$.

Q: What is the largest value of x that satisfies the equation $\log_3\left(x^2\right) - \log_3(x+2) = 1$?

A: The largest value of $x$ that satisfies the equation $\log_3\left(x^2\right) - \log_3(x+2) = 1$ is $x = 6$, but this is not a solution to the original equation. Therefore, the largest value of $x$ that satisfies the equation is actually $x = \boxed{6}$.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a computer program. You can also use a table of values to graph the function. For example, if you have the function $f(x) = \log_3(x)$, you can use a table of values to graph the function.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers greater than zero. For example, the function $f(x) = \log_3(x)$ is defined for all real numbers greater than zero.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is all real numbers. For example, the function $f(x) = \log_3(x)$ has a range of all real numbers.

Conclusion

In this article, we answered some common questions that readers may have about solving logarithmic equations. We also provided some tips and tricks for simplifying and solving logarithmic equations. We hope that this article has been helpful in understanding logarithmic equations and how to solve them.

Additional Resources

If you are interested in learning more about logarithmic equations, we recommend checking out the following resources:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

These resources provide a wealth of information on logarithmic equations, including tutorials, examples, and practice problems.

Final Thoughts

Logarithmic equations can be challenging to solve, but with practice and patience, you can become proficient in solving them. Remember to use the properties of logarithms to simplify the equation and then solve for the variable. With these tips and tricks, you will be well on your way to becoming a logarithmic equation expert!