What Is The Solution To Log ⁡ 4 ( 8 X ) = 3 \log_4(8x) = 3 Lo G 4 ​ ( 8 X ) = 3 ?A. X = 3 8 X = \frac{3}{8} X = 8 3 ​ B. X = 3 2 X = \frac{3}{2} X = 2 3 ​ C. X = 2 X = 2 X = 2 D. X = 8 X = 8 X = 8

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Introduction

In this article, we will explore the solution to the logarithmic equation $\log_4(8x) = 3$. This equation involves a logarithm with base 4, and we need to isolate the variable $x$ to find its value. We will use the properties of logarithms and algebraic manipulations to solve this equation.

Understanding Logarithms

Before we dive into the solution, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a(c) = b$. The logarithm with base $a$ of a number $c$ is the exponent to which $a$ must be raised to produce $c$.

The Given Equation

The given equation is $\log_4(8x) = 3$. This equation states that the logarithm with base 4 of $8x$ is equal to 3. We can rewrite this equation in exponential form as $4^3 = 8x$.

Solving the Equation

Now, let's solve the equation $4^3 = 8x$. We know that $4^3 = 64$, so we can rewrite the equation as $64 = 8x$. To isolate $x$, we need to divide both sides of the equation by 8.

# Import necessary modules
import math
a = 64
b = 8

x = a / b
print(x)

Calculating the Value of $x$

When we divide 64 by 8, we get $x = 8$. Therefore, the solution to the equation $\log_4(8x) = 3$ is $x = 8$.

Conclusion

In this article, we have solved the logarithmic equation $\log_4(8x) = 3$. We used the properties of logarithms and algebraic manipulations to isolate the variable $x$ and find its value. The solution to this equation is $x = 8$.

Final Answer

The final answer is $x = 8$. This corresponds to option D in the given multiple-choice question.

Discussion

The solution to this equation involves understanding the properties of logarithms and algebraic manipulations. It is essential to review the concept of logarithms and practice solving logarithmic equations to become proficient in this area of mathematics.

Related Topics

• Logarithmic equations
• Algebraic manipulations
• Properties of logarithms

References

• [1] "Logarithms" by Khan Academy
• [2] "Algebraic Manipulations" by Math Open Reference
• [3] "Properties of Logarithms" by Wolfram MathWorld
  

Introduction

In the previous article, we explored the solution to the logarithmic equation $\log_4(8x) = 3$. In this article, we will address some frequently asked questions (FAQs) about logarithmic equations. These FAQs will provide additional insights and clarify common misconceptions about 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, whereas an exponential equation is an equation that involves an exponent. For example, $\log_4(8x) = 3$ is a logarithmic equation, while $4^3 = 8x$ is an exponential equation.

Q: How do I solve a logarithmic equation with a base other than 10?

A: To solve a logarithmic equation with a base other than 10, you can use the change of base formula. The change of base formula states that $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$, where $a$, $b$, and $c$ are positive real numbers.

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

A: Yes, you can use a calculator to solve a logarithmic equation. However, it's essential to understand the concept behind the calculation and to verify the solution using algebraic manipulations.

Q: How do I determine the domain of a logarithmic function?

A: The domain of a logarithmic function is the set of all possible input values for which the function is defined. For a logarithmic function $\log_a(x)$, the domain is all positive real numbers.

Q: Can I use logarithmic equations to solve exponential equations?

A: Yes, you can use logarithmic equations to solve exponential equations. By taking the logarithm of both sides of an exponential equation, you can rewrite it as a logarithmic equation, which can then be solved using logarithmic properties.

Q: What is the relationship between logarithmic and exponential functions?

A: Logarithmic and exponential functions are inverse functions. This means that if $f(x) = a^x$ is an exponential function, then $f^{-1}(x) = \log_a(x)$ is a logarithmic function.

Q: Can I use logarithmic equations to model real-world problems?

A: Yes, logarithmic equations can be used to model real-world problems involving growth and decay, such as population growth, chemical reactions, and financial investments.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or software. Alternatively, you can use the properties of logarithmic functions to determine the key features of the graph, such as the domain, range, and asymptotes.

Q: Can I use logarithmic equations to solve systems of equations?

A: Yes, you can use logarithmic equations to solve systems of equations. By taking the logarithm of both sides of one or more equations, you can rewrite the system as a set of logarithmic equations, which can then be solved using logarithmic properties.

Conclusion

In this article, we have addressed some frequently asked questions about logarithmic equations. These FAQs provide additional insights and clarify common misconceptions about logarithmic equations. By understanding the properties and applications of logarithmic equations, you can become proficient in solving logarithmic equations and apply them to real-world problems.

Final Answer

The final answer is that logarithmic equations are a powerful tool for solving problems involving growth and decay, and they can be used to model a wide range of real-world phenomena.

Discussion

The discussion of logarithmic equations is an ongoing process, and there is always more to learn and explore. By continuing to study and practice logarithmic equations, you can develop a deeper understanding of their properties and applications.

Related Topics

• Exponential equations
• Logarithmic properties
• Graphing logarithmic functions
• Systems of equations

References

• [1] "Logarithmic Equations" by Khan Academy
• [2] "Exponential Equations" by Math Open Reference
• [3] "Graphing Logarithmic Functions" by Wolfram MathWorld