Solve The Following Logarithmic Equation. Be Sure To Reject Any Value Of X X X That Is Not In The Domain Of The Original Logarithmic Expression. Give The Exact Answer. Log ⁡ 3 ( X + 6 ) = 4 \log_3(x+6)=4 Lo G 3 ​ ( X + 6 ) = 4 Solve The Equation. Select The Correct Choice Below

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Introduction

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Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will focus on solving the logarithmic equation $\log_3(x+6)=4$. We will use the properties of logarithms to isolate the variable $x$ and find the exact solution.

Understanding the Domain of the Logarithmic Expression

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Before we start solving the equation, it's essential to understand the domain of the logarithmic expression. The domain of a logarithmic expression is the set of all possible values of the input that can be plugged into the logarithm. In this case, the input is $x+6$, and the base of the logarithm is $3$. Since the logarithm is defined only for positive real numbers, we must have $x+6>0$. This implies that $x>-6$.

Using the Definition of a Logarithm to Rewrite the Equation

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The definition of a logarithm states that if $y=\log_b(x)$, then $b^y=x$. We can use this definition to rewrite the equation $\log_3(x+6)=4$ as an exponential equation. By raising both sides of the equation to the power of $3$, we get:

$$3^{\log_3(x+6)}=3^4$$

Using the property of logarithms that states $\log_b(b^x)=x$, we can simplify the left-hand side of the equation to get:

$$x+6=81$$

Solving for $x$

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Now that we have a linear equation, we can solve for $x$ by subtracting $6$ from both sides of the equation:

$$x=81-6$$

$$x=75$$

Checking the Solution

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Before we conclude that $x=75$ is the solution to the equation, we need to check that it satisfies the original equation. We can do this by plugging $x=75$ back into the original equation:

$$\log_3(75+6)=\log_3(81)=4$$

Since this is true, we can conclude that $x=75$ is indeed the solution to the equation.

Conclusion

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In this article, we solved the logarithmic equation $\log_3(x+6)=4$ using the properties of logarithms. We first understood the domain of the logarithmic expression and then used the definition of a logarithm to rewrite the equation as an exponential equation. Finally, we solved for $x$ and checked that the solution satisfied the original equation. The exact solution to the equation is $x=75$.

Frequently Asked Questions

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Q: What is the domain of the logarithmic expression $\log_3(x+6)$?

A: The domain of the logarithmic expression $\log_3(x+6)$ is all real numbers $x$ such that $x+6>0$, which implies $x>-6$.

Q: How do you rewrite a logarithmic equation as an exponential equation?

A: To rewrite a logarithmic equation as an exponential equation, you can raise both sides of the equation to the power of the base of the logarithm.

Q: How do you solve a linear equation?

A: To solve a linear equation, you can add or subtract the same value from both sides of the equation to isolate the variable.

Final Answer

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The final answer is: $\boxed{75}$

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Introduction

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Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will provide a comprehensive Q&A guide to help you understand and solve logarithmic equations.

Q&A

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Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. It is an equation that can be written in the form $\log_b(x)=y$, where $b$ is the base of the logarithm, $x$ is the input, and $y$ is the output.

Q: What is the domain of a logarithmic expression?

A: The domain of a logarithmic expression is the set of all possible values of the input that can be plugged into the logarithm. For a logarithmic expression $\log_b(x)$, the domain is all real numbers $x$ such that $x>0$.

Q: How do you rewrite a logarithmic equation as an exponential equation?

A: To rewrite a logarithmic equation as an exponential equation, you can raise both sides of the equation to the power of the base of the logarithm. For example, if you have the equation $\log_3(x)=4$, you can rewrite it as $3^4=x$.

Q: How do you solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to rewrite the equation as an exponential equation, and then solve for the variable. For example, if you have the equation $\log_3(x+6)=4$, you can rewrite it as $3^4=x+6$, and then solve for $x$.

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 exponent. For example, the equation $\log_3(x)=4$ is a logarithmic equation, while the equation $3^4=x$ is an exponential equation.

Q: How do you check the solution to a logarithmic equation?

A: To check the solution to a logarithmic equation, you can plug the solution back into the original equation and verify that it is true. For example, if you have the equation $\log_3(x+6)=4$ and you find that $x=75$ is a solution, you can plug $x=75$ back into the original equation to verify that it is 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 checking the domain of the logarithmic expression
• Not rewriting the logarithmic equation as an exponential equation
• Not solving for the variable correctly
• Not checking the solution to the equation

Tips and Tricks

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Tip 1: Understand the properties of logarithms

To solve logarithmic equations, it's essential to understand the properties of logarithms, including the definition of a logarithm, the properties of logarithms, and how to rewrite logarithmic equations as exponential equations.

Tip 2: Use the definition of a logarithm to rewrite the equation

To rewrite a logarithmic equation as an exponential equation, you can use the definition of a logarithm, which states that if $y=\log_b(x)$, then $b^y=x$.

Tip 3: Check the solution to the equation

To ensure that you have found the correct solution to the equation, you should check the solution by plugging it back into the original equation.

Conclusion

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In this article, we provided a comprehensive Q&A guide to help you understand and solve logarithmic equations. We covered topics such as the definition of a logarithmic equation, the domain of a logarithmic expression, and how to rewrite a logarithmic equation as an exponential equation. We also provided tips and tricks to help you avoid common mistakes when solving logarithmic equations.

Final Answer

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The final answer is: $\boxed{75}$