What Is The Solution To $\log_4 8x = 3$?A. $x = \frac{3}{8}$B. $x = \frac{3}{2}$C. $x = 2$D. $x = 8$

Understanding the Problem

The problem involves solving a logarithmic equation, specifically $\log_4 8x = 3$. To solve this equation, we need to understand the properties of logarithms and how to manipulate them to isolate the variable $x$. Logarithms are the inverse operation of exponentiation, and they play a crucial role in mathematics, particularly in solving equations involving exponential functions.

Properties of Logarithms

Before we dive into solving the equation, let's review some essential properties of logarithms. The logarithm of a number $a$ with base $b$ is denoted as $\log_b a$. The product rule of logarithms states that $\log_b (xy) = \log_b x + \log_b y$. The power rule of logarithms states that $\log_b (x^y) = y \log_b x$. These properties will be useful in solving the equation.

Solving the Equation

Now, let's focus on solving the equation $\log_4 8x = 3$. To start, we can rewrite the equation in exponential form. Since the base of the logarithm is 4, we can rewrite the equation as $4^3 = 8x$. This simplifies to $64 = 8x$.

Isolating the Variable

Next, we need to isolate the variable $x$. To do this, we can divide both sides of the equation by 8. This gives us $x = \frac{64}{8}$.

Simplifying the Expression

Now, let's simplify the expression $\frac{64}{8}$. We can do this by dividing the numerator and denominator by their greatest common divisor, which is 8. This gives us $x = \frac{64 \div 8}{8 \div 8} = \frac{8}{1}$.

Evaluating the Expression

Finally, let's evaluate the expression $\frac{8}{1}$. This simplifies to $x = 8$.

Conclusion

In conclusion, the solution to the equation $\log_4 8x = 3$ is $x = 8$. This is the only option that satisfies the equation.

Comparison with Options

Let's compare our solution with the options provided:

• A. $x = \frac{3}{8}$: This is not the correct solution.
• B. $x = \frac{3}{2}$: This is not the correct solution.
• C. $x = 2$: This is not the correct solution.
• D. $x = 8$: This is the correct solution.

Final Answer

The final answer is $\boxed{8}$.

Additional Tips and Tricks

Here are some additional tips and tricks for solving logarithmic equations:

• Always rewrite the equation in exponential form.
• Use the product rule and power rule of logarithms to simplify the equation.
• Isolate the variable by dividing both sides of the equation by the coefficient of the variable.
• Simplify the expression by dividing the numerator and denominator by their greatest common divisor.
• Evaluate the expression to find the final answer.

By following these tips and tricks, you can solve logarithmic equations with ease and confidence.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. Logarithmic equations are used to solve problems that involve exponential functions.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to rewrite the equation in exponential form, use the product rule and power rule of logarithms to simplify the equation, isolate the variable by dividing both sides of the equation by the coefficient of the variable, simplify the expression by dividing the numerator and denominator by their greatest common divisor, and evaluate the expression to find the final answer.

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 rewrite a logarithmic equation in exponential form?

A: To rewrite a logarithmic equation in exponential form, you need to use the definition of a logarithm. For example, the equation $\log_4 8x = 3$ can be rewritten in exponential form as $4^3 = 8x$.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_b (xy) = \log_b x + \log_b y$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_b (x^y) = y \log_b x$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you need to use the product rule and power rule of logarithms to combine the logarithms of the individual factors. For example, the expression $\log_4 (8x)$ can be simplified as $\log_4 8 + \log_4 x$.

Q: What is the difference between a base-10 logarithm and a base-4 logarithm?

A: A base-10 logarithm is a logarithm with a base of 10, while a base-4 logarithm is a logarithm with a base of 4. For example, the equation $\log_{10} 100 = 2$ is a base-10 logarithm, while the equation $\log_4 64 = 3$ is a base-4 logarithm.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to use the definition of a logarithm. For example, the expression $\log_4 64$ can be evaluated as $3$.

Q: What are some common applications of logarithmic equations?

A: Logarithmic equations have many applications in mathematics, science, and engineering. Some common applications include:

• Solving problems involving exponential growth and decay
• Modeling population growth and decline
• Analyzing data and making predictions
• Solving problems involving finance and economics
• Solving problems involving physics and engineering

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

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

• Not rewriting the equation in exponential form
• Not using the product rule and power rule of logarithms to simplify the equation
• Not isolating the variable by dividing both sides of the equation by the coefficient of the variable
• Not simplifying the expression by dividing the numerator and denominator by their greatest common divisor
• Not evaluating the expression to find the final answer.

By following these tips and avoiding common mistakes, you can solve logarithmic equations with ease and confidence.