Select The Correct Answer.Find The Value Of $x$.$\log _x 8 = 0.5$A. 16 B. 64 C. 32 D. 4

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Introduction

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Logarithmic equations can be challenging to solve, but with a clear understanding of the properties of logarithms, we can break them down into manageable steps. In this article, we will focus on solving the equation $\log _x 8 = 0.5$ to find the value of $x$. We will explore the properties of logarithms, apply them to the given equation, and arrive at the correct solution.

Understanding Logarithmic Equations

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A logarithmic equation is an equation that involves a logarithm. The general form of a logarithmic equation is $\log _a b = c$, where $a$ is the base of the logarithm, $b$ is the argument of the logarithm, and $c$ is the result of the logarithm. In the given equation, $\log _x 8 = 0.5$, $x$ is the base, $8$ is the argument, and $0.5$ is the result.

Properties of Logarithms

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To solve the given equation, we need to understand the properties of logarithms. The two main properties of logarithms are:

• Product Property: $\log _a (b \cdot c) = \log _a b + \log _a c$
• Power Property: $\log _a b^c = c \cdot \log _a b$

Applying the Properties of Logarithms

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Now that we have a good understanding of the properties of logarithms, we can apply them to the given equation. We can start by rewriting the equation in exponential form:

$$x^{0.5} = 8$$

Solving for $x$

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To solve for $x$, we can raise both sides of the equation to the power of $2$:

$$x = 8^2$$

Calculating the Value of $x$

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Now that we have the equation $x = 8^2$, we can calculate the value of $x$:

$$x = 64$$

Conclusion

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In this article, we have solved the logarithmic equation $\log _x 8 = 0.5$ to find the value of $x$. We have applied the properties of logarithms, rewritten the equation in exponential form, and solved for $x$. The correct answer is $\boxed{64}$.

Frequently Asked Questions

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Q: What is the base of the logarithm in the given equation?

A: The base of the logarithm in the given equation is $x$.

Q: What is the argument of the logarithm in the given equation?

A: The argument of the logarithm in the given equation is $8$.

Q: What is the result of the logarithm in the given equation?

A: The result of the logarithm in the given equation is $0.5$.

Q: How do we solve the given equation?

A: We solve the given equation by applying the properties of logarithms, rewriting the equation in exponential form, and solving for $x$.

Q: What is the value of $x$ in the given equation?

A: The value of $x$ in the given equation is $64$.

Additional Resources

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For more information on logarithmic equations and how to solve them, check out the following resources:

• Logarithmic Equations
• Solving Logarithmic Equations
• Logarithmic Equations and Inequalities

Final Thoughts

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Solving logarithmic equations requires a clear understanding of the properties of logarithms and how to apply them to the given equation. By following the steps outlined in this article, we can solve even the most challenging logarithmic equations. Remember to always apply the properties of logarithms, rewrite the equation in exponential form, and solve for the variable. With practice and patience, you will become proficient in solving logarithmic equations and be able to tackle even the most complex problems.

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Introduction

---

Logarithmic equations can be challenging to solve, but with a clear understanding of the properties of logarithms, we can break them down into manageable steps. In this article, we will answer some of the most frequently asked questions about logarithmic equations, providing a comprehensive guide to help you understand and solve these types of equations.

Q: What is a logarithmic equation?

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A logarithmic equation is an equation that involves a logarithm. The general form of a logarithmic equation is $\log _a b = c$, where $a$ is the base of the logarithm, $b$ is the argument of the logarithm, and $c$ is the result of the logarithm.

Q: How do I solve a logarithmic equation?

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To solve a logarithmic equation, you need to apply the properties of logarithms, rewrite the equation in exponential form, and solve for the variable. Here's a step-by-step guide:

1. Apply the properties of logarithms: Use the product property and power property to simplify the equation.
2. Rewrite the equation in exponential form: Use the definition of a logarithm to rewrite the equation in exponential form.
3. Solve for the variable: Use algebraic techniques to solve for the variable.

Q: What is the base of the logarithm?

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The base of the logarithm is the number that is used to raise the argument to the power of the result. For example, in the equation $\log _x 8 = 0.5$, the base is $x$.

Q: What is the argument of the logarithm?

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The argument of the logarithm is the number that is being raised to the power of the result. For example, in the equation $\log _x 8 = 0.5$, the argument is $8$.

Q: What is the result of the logarithm?

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The result of the logarithm is the exponent that is used to raise the base to the power of the argument. For example, in the equation $\log _x 8 = 0.5$, the result is $0.5$.

Q: How do I convert a logarithmic equation to exponential form?

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To convert a logarithmic equation to exponential form, you need to use the definition of a logarithm. The definition of a logarithm is:

$$\log _a b = c \iff a^c = b$$

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

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A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example:

• Logarithmic equation: $\log _x 8 = 0.5$
• Exponential equation: $x^0.5 = 8$

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

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Yes, you can use a calculator to solve a logarithmic equation. However, it's always a good idea to understand the underlying math and apply the properties of logarithms to simplify the equation before using a calculator.

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

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Here are some common mistakes to avoid when solving logarithmic equations:

• Not applying the properties of logarithms: Make sure to apply the product property and power property to simplify the equation.
• Not rewriting the equation in exponential form: Use the definition of a logarithm to rewrite the equation in exponential form.
• Not solving for the variable: Use algebraic techniques to solve for the variable.

Conclusion

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Logarithmic equations can be challenging to solve, but with a clear understanding of the properties of logarithms and a step-by-step approach, you can break them down into manageable steps. Remember to apply the properties of logarithms, rewrite the equation in exponential form, and solve for the variable. With practice and patience, you will become proficient in solving logarithmic equations and be able to tackle even the most complex problems.

Additional Resources

---

For more information on logarithmic equations and how to solve them, check out the following resources:

• Logarithmic Equations
• Solving Logarithmic Equations
• Logarithmic Equations and Inequalities

Final Thoughts

---

Solving logarithmic equations requires a clear understanding of the properties of logarithms and how to apply them to the given equation. By following the steps outlined in this article, you can solve even the most challenging logarithmic equations. Remember to always apply the properties of logarithms, rewrite the equation in exponential form, and solve for the variable. With practice and patience, you will become proficient in solving logarithmic equations and be able to tackle even the most complex problems.