Solve The Following Logarithmic Equation. Be Sure To Reject Any Value Of $x$ That Is Not In The Domain Of The Original Logarithmic Expression. Give The Exact Answer. Log ⁡ 3 X = 4 \log _3 X = 4 Lo G 3 ​ X = 4 Solve The Equation. Select The Correct Choice Below

Introduction

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 $\log _3 x = 4$. We will use the properties of logarithms to isolate the variable $x$ and find its exact value.

Understanding the Domain of the Logarithmic Expression

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 variable that can be input into the expression without resulting in a non-real or undefined value.

In this case, the logarithmic expression is $\log _3 x$. The base of the logarithm is 3, and the variable is $x$. The domain of this expression is all real numbers greater than 0, since the logarithm of a non-positive number is undefined.

Solving the Logarithmic Equation

Now that we have a clear understanding of the domain of the logarithmic expression, we can start solving the equation.

The given equation is $\log _3 x = 4$. To solve for $x$, we can use the property of logarithms that states $\log _a b = c \iff a^c = b$.

Applying this property to the given equation, we get:

$$3^4 = x$$

Simplifying the Equation

Now that we have isolated the variable $x$, we can simplify the equation by evaluating the exponent.

$$3^4 = 81$$

Therefore, the solution to the equation is $x = 81$.

Checking the Solution

Before we conclude that $x = 81$ is the only solution to the equation, we need to check that it is indeed in the domain of the original logarithmic expression.

Since $x = 81$ is a positive real number, it is in the domain of the logarithmic expression. Therefore, $x = 81$ is a valid solution to the equation.

Conclusion

In this article, we solved the logarithmic equation $\log _3 x = 4$ using the properties of logarithms. We found that the solution to the equation is $x = 81$, and we verified that it is in the domain of the original logarithmic expression.

Key Takeaways

• The domain of a logarithmic expression is all real numbers greater than 0.
• To solve a logarithmic equation, use the property of logarithms that states $\log _a b = c \iff a^c = b$.
• Always check that the solution to a logarithmic equation is in the domain of the original logarithmic expression.

Final Answer

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Introduction

In our previous article, we solved the logarithmic equation $\log _3 x = 4$ using the properties of logarithms. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving logarithmic equations.

Q: What is the domain of a logarithmic expression?

A: The domain of a logarithmic expression is all real numbers greater than 0. This is because the logarithm of a non-positive number is undefined.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, use the property of logarithms that states $\log _a b = c \iff a^c = b$. This property allows you to rewrite the logarithmic equation in exponential form and solve for the variable.

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

A: A logarithmic equation is an equation that involves a logarithmic expression, such as $\log _a b = c$. An exponential equation, on the other hand, is an equation that involves an exponential expression, such as $a^c = b$.

Q: How do I check if a solution to a logarithmic equation is in the domain of the original logarithmic expression?

A: To check if a solution to a logarithmic equation is in the domain of the original logarithmic expression, simply verify that the solution is a positive real number.

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 if the solution is in the domain of the original logarithmic expression
• Not using the correct property of logarithms to rewrite the equation in exponential form
• Not simplifying the equation correctly

Q: Can you provide an example of a logarithmic equation that involves a base other than 10?

A: Yes, here is an example of a logarithmic equation that involves a base other than 10:

$$\log _5 x = 3$$

To solve this equation, use the property of logarithms that states $\log _a b = c \iff a^c = b$. This gives:

$$5^3 = x$$

Simplifying the equation, we get:

$$x = 125$$

Q: Can you provide an example of a logarithmic equation that involves a base of 2?

A: Yes, here is an example of a logarithmic equation that involves a base of 2:

$$\log _2 x = 5$$

To solve this equation, use the property of logarithms that states $\log _a b = c \iff a^c = b$. This gives:

$$2^5 = x$$

Simplifying the equation, we get:

$$x = 32$$

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving logarithmic equations. We covered topics such as the domain of a logarithmic expression, how to solve a logarithmic equation, and common mistakes to avoid.

Key Takeaways

• The domain of a logarithmic expression is all real numbers greater than 0.
• To solve a logarithmic equation, use the property of logarithms that states $\log _a b = c \iff a^c = b$.
• Always check if the solution to a logarithmic equation is in the domain of the original logarithmic expression.

Final Answer

The final answer is $\boxed{32}$.