Select The Correct Answer From The Drop-down Menu.For A Certain Base, B B B , Log ⁡ B 8 = 3 \log_b 8 = 3 Lo G B ​ 8 = 3 And Log ⁡ B 0.5 = − 1 \log_b 0.5 = -1 Lo G B ​ 0.5 = − 1 . The Value Of Log ⁡ B 4 B \log_b 4b Lo G B ​ 4 B Is □ \square □ .

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 explore how to solve logarithmic equations using the given information about a certain base, $b$, where $\log_b 8 = 3$ and $\log_b 0.5 = -1$. We will then use this information to find the value of $\log_b 4b$.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $x = \log_b y$, then $b^x = y$. The logarithmic equation $\log_b 8 = 3$ can be rewritten as $b^3 = 8$, which means that the base $b$ raised to the power of 3 is equal to 8.

Using the Change of Base Formula

The change of base formula is a useful tool for solving logarithmic equations. It states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$, $b$, and $c$ are positive real numbers. We can use this formula to rewrite the given logarithmic equations in terms of a common base.

Rewriting the Logarithmic Equations

Using the change of base formula, we can rewrite the logarithmic equations as follows:

• $\log_b 8 = 3 \Rightarrow \frac{\log 8}{\log b} = 3$
• $\log_b 0.5 = -1 \Rightarrow \frac{\log 0.5}{\log b} = -1$

Simplifying the Logarithmic Equations

We can simplify the logarithmic equations by using the properties of logarithms. Specifically, we can use the fact that $\log a^b = b \log a$.

• $\frac{\log 8}{\log b} = 3 \Rightarrow \log 8 = 3 \log b$
• $\frac{\log 0.5}{\log b} = -1 \Rightarrow \log 0.5 = -\log b$

Finding the Value of $\log_b 4b$

Now that we have simplified the logarithmic equations, we can use them to find the value of $\log_b 4b$. We can start by rewriting the expression $\log_b 4b$ as follows:

• $\log_b 4b = \log_b (4 \cdot b)$

Using the Product Rule

The product rule states that $\log_b (a \cdot c) = \log_b a + \log_b c$. We can use this rule to rewrite the expression $\log_b 4b$ as follows:

• $\log_b 4b = \log_b 4 + \log_b b$

Finding the Value of $\log_b 4$

We can find the value of $\log_b 4$ by using the fact that $\log_b 8 = 3$. Specifically, we can rewrite the expression $\log_b 8$ as follows:

• $\log_b 8 = \log_b (2^3) = 3 \log_b 2$

Finding the Value of $\log_b 2$

We can find the value of $\log_b 2$ by using the fact that $\log_b 0.5 = -1$. Specifically, we can rewrite the expression $\log_b 0.5$ as follows:

• $\log_b 0.5 = \log_b (2^{-1}) = -\log_b 2$

Solving for $\log_b 2$

We can solve for $\log_b 2$ by equating the two expressions:

• $-\log_b 2 = -1 \Rightarrow \log_b 2 = 1$

Finding the Value of $\log_b 4$

Now that we have found the value of $\log_b 2$, we can find the value of $\log_b 4$:

• $\log_b 4 = 2 \log_b 2 = 2 \cdot 1 = 2$

Finding the Value of $\log_b 4b$

Finally, we can find the value of $\log_b 4b$ by substituting the values of $\log_b 4$ and $\log_b b$:

• $\log_b 4b = \log_b 4 + \log_b b = 2 + 1 = 3$

Conclusion

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Introduction

In our previous article, we explored how to solve logarithmic equations using the given information about a certain base, $b$, where $\log_b 8 = 3$ and $\log_b 0.5 = -1$. We used the change of base formula, the product rule, and the properties of logarithms to simplify the logarithmic equations and find the value of $\log_b 4b$. In this article, we will provide a Q&A guide to help you better understand the concepts and solve logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $x = \log_b y$, then $b^x = y$.

Q: How do I rewrite a logarithmic equation using the change of base formula?

A: To rewrite a logarithmic equation using the change of base formula, you can use the following formula:

$\log_b a = \frac{\log_c a}{\log_c b}$

where $a$, $b$, and $c$ are positive real numbers.

Q: What is the product rule for logarithms?

A: The product rule for logarithms states that:

$\log_b (a \cdot c) = \log_b a + \log_b c$

This rule allows you to simplify logarithmic expressions by combining the logarithms of multiple terms.

Q: How do I find the value of $\log_b 4b$?

A: To find the value of $\log_b 4b$, you can use the product rule and the properties of logarithms. Specifically, you can rewrite the expression $\log_b 4b$ as:

$\log_b 4b = \log_b 4 + \log_b b$

Then, you can use the fact that $\log_b 8 = 3$ and $\log_b 0.5 = -1$ to find the values of $\log_b 4$ and $\log_b b$.

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. In other words, if $x = \log_b y$, then $b^x = y$. Exponential equations are the inverse of logarithmic equations.

Q: How do I solve a logarithmic equation with a variable base?

A: To solve a logarithmic equation with a variable base, you can use the change of base formula and the properties of logarithms. Specifically, you can rewrite the equation in terms of a common base and then use algebraic techniques to solve for the variable.

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

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

• Forgetting to use the change of base formula
• Not using the product rule for logarithms
• Not checking the domain of the logarithmic function
• Not using algebraic techniques to solve for the variable

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concepts and solve logarithmic equations. We have covered topics such as the change of base formula, the product rule, and the properties of logarithms. We have also provided examples and explanations to help you understand the concepts. By following these guidelines and practicing with examples, you will become more confident and proficient in solving logarithmic equations.

Additional Resources

• Khan Academy: Logarithms
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

Practice Problems

1. Solve the logarithmic equation $\log_b 16 = 4$.
2. Solve the logarithmic equation $\log_b 0.25 = -2$.
3. Find the value of $\log_b 6b$.
4. Solve the logarithmic equation $\log_b 9 = 2$.
5. Find the value of $\log_b 3b$.

Answer Key

1. $b = 2$
2. $b = 0.5$
3. $3$
4. $b = 3$
5. $2$