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

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be broken down into manageable steps. In this article, we will focus on solving the equation $\log_4 8x = 3$. This equation involves a logarithm with a base of 4 and an argument of $8x$. Our goal is to isolate the variable $x$ and find its value.

Understanding Logarithms

Before we dive into solving the equation, let's take a moment to understand what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, if we have an equation of the form $a^b = c$, then the logarithm of $c$ with base $a$ is equal to $b$. This can be written as $\log_a c = b$.

The Equation $\log_4 8x = 3$

Now that we have a basic understanding of logarithms, let's focus on the equation at hand. We are given the equation $\log_4 8x = 3$. Our goal is to isolate the variable $x$ and find its value.

Step 1: Rewrite the Equation

The first step in solving this equation is to rewrite it in exponential form. We can do this by using the definition of a logarithm. Since $\log_4 8x = 3$, we can rewrite this as $4^3 = 8x$.

Step 2: Simplify the Equation

Now that we have rewritten the equation in exponential form, let's simplify it. We can start by evaluating the left-hand side of the equation. Since $4^3 = 64$, we can rewrite the equation as $64 = 8x$.

Step 3: Isolate the Variable $x$

Now that we have simplified the equation, let's isolate the variable $x$. We can do this by dividing both sides of the equation by 8. This gives us $x = \frac{64}{8}$.

Step 4: Simplify the Fraction

Now that we have isolated the variable $x$, let's simplify the fraction. We can do this by dividing both the numerator and the denominator by their greatest common divisor, which is 8. This gives us $x = \frac{8}{1}$.

Step 5: Evaluate the Expression

Now that we have simplified the fraction, let's evaluate the expression. We can do this by dividing 8 by 1, which gives us $x = 8$.

Conclusion

In this article, we have solved the equation $\log_4 8x = 3$ using a step-by-step approach. We started by rewriting the equation in exponential form, then simplified it, isolated the variable $x$, simplified the fraction, and finally evaluated the expression. Our final answer is $x = 8$.

Answer Key

The correct answer is D. $x = 8$.

Additional Tips and Tricks

• When solving logarithmic equations, it's essential to remember that the base of the logarithm is the same as the base of the exponential function.
• When rewriting logarithmic equations in exponential form, make sure to use the correct base.
• When simplifying fractions, make sure to divide both the numerator and the denominator by their greatest common divisor.
• When evaluating expressions, make sure to follow the order of operations (PEMDAS).

Common Mistakes to Avoid

• When solving logarithmic equations, it's easy to get confused between the base of the logarithm and the base of the exponential function. Make sure to double-check your work to avoid this mistake.
• When rewriting logarithmic equations in exponential form, make sure to use the correct base. If you use the wrong base, you may end up with an incorrect solution.
• When simplifying fractions, make sure to divide both the numerator and the denominator by their greatest common divisor. If you don't do this, you may end up with an incorrect solution.
• When evaluating expressions, make sure to follow the order of operations (PEMDAS). If you don't do this, you may end up with an incorrect solution.

Real-World Applications

Logarithmic equations have many real-world applications. For example, they can be used to model population growth, chemical reactions, and financial transactions. In addition, logarithmic equations can be used to solve problems in physics, engineering, and computer science.

Conclusion

Q: What is the definition of a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if we have an equation of the form $a^b = c$, then the logarithm of $c$ with base $a$ is equal to $b$. This can be written as $\log_a c = b$.

Q: How do I rewrite a logarithmic equation in exponential form?

A: To rewrite a logarithmic equation in exponential form, you can use the definition of a logarithm. For example, if we have the equation $\log_4 8x = 3$, we can rewrite it in exponential form as $4^3 = 8x$.

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 $\log_4 8x = 3$ is a logarithmic equation, while the equation $4^3 = 8x$ is an exponential equation.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can start by rewriting it in exponential form. Then, you can simplify the resulting equation by combining like terms and isolating the variable.

Q: What is the order of operations for logarithmic equations?

A: The order of operations for logarithmic equations is the same as for exponential equations: parentheses, exponents, multiplication and division, and addition and subtraction.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can start by rewriting it in exponential form. Then, you can evaluate the resulting expression by following the order of operations.

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

A: Some common mistakes to avoid when solving logarithmic equations include:

• Getting confused between the base of the logarithm and the base of the exponential function
• Using the wrong base when rewriting a logarithmic equation in exponential form
• Failing to simplify the resulting equation
• Failing to follow the order of operations

Q: What are some real-world applications of logarithmic equations?

A: Logarithmic equations have many real-world applications, including:

• Modeling population growth
• Solving chemical reactions
• Analyzing financial transactions
• Solving problems in physics, engineering, and computer science

Q: How can I practice solving logarithmic equations?

A: You can practice solving logarithmic equations by working through example problems and exercises. You can also try solving real-world problems that involve logarithmic equations.

Q: What are some resources for learning more about logarithmic equations?

A: Some resources for learning more about logarithmic equations include:

• Online tutorials and videos
• Textbooks and workbooks
• Online forums and communities
• Math classes and workshops

Conclusion

In conclusion, logarithmic equations are an important topic in mathematics that have many real-world applications. By understanding the definition of a logarithm, how to rewrite logarithmic equations in exponential form, and how to simplify and evaluate logarithmic expressions, you can solve even the most challenging logarithmic equations. Remember to always double-check your work and follow the order of operations to avoid common mistakes. With practice and patience, you can become proficient in solving logarithmic equations and apply them to real-world problems.