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 $b^x = y$. Logarithmic equations can be solved using various properties of logarithms, including the product rule, the quotient rule, and the power rule.

The Product Rule

The product rule states that $\log_b (xy) = \log_b x + \log_b y$. This rule can be used to simplify logarithmic expressions and solve logarithmic equations.

The Quotient Rule

The quotient rule states that $\log_b (x/y) = \log_b x - \log_b y$. This rule can be used to simplify logarithmic expressions and solve logarithmic equations.

The Power Rule

The power rule states that $\log_b x^y = y \log_b x$. This rule can be used to simplify logarithmic expressions and solve logarithmic equations.

Solving the Given Equations

We are given two logarithmic equations:

1. $\log_b 8 = 3$
2. $\log_b 0.5 = -1$

We can use these equations to find the value of $\log_b 4b$.

Using the Product Rule

We can rewrite the equation $\log_b 8 = 3$ as $\log_b (2^3) = 3$. Using the product rule, we can rewrite this equation as $\log_b 2 + \log_b 2 + \log_b 2 = 3$. Simplifying this equation, we get $3 \log_b 2 = 3$.

Solving for $\log_b 2$

We can solve for $\log_b 2$ by dividing both sides of the equation by 3. This gives us $\log_b 2 = 1$.

Using the Quotient Rule

We can rewrite the equation $\log_b 0.5 = -1$ as $\log_b (2^{-1}) = -1$. Using the quotient rule, we can rewrite this equation as $\log_b 2 - \log_b 2 = -1$. Simplifying this equation, we get $0 = -1$.

Solving for $\log_b 2$

We can solve for $\log_b 2$ by adding 1 to both sides of the equation. This gives us $\log_b 2 = 1$.

Finding the Value of $\log_b 4b$

We can use the product rule to rewrite the expression $\log_b 4b$ as $\log_b (2^2) + \log_b b$. Using the power rule, we can rewrite this expression as $2 \log_b 2 + \log_b b$.

Substituting the Value of $\log_b 2$

We can substitute the value of $\log_b 2$ into the expression $2 \log_b 2 + \log_b b$. This gives us $2(1) + \log_b b$.

Simplifying the Expression

We can simplify the expression $2(1) + \log_b b$ by evaluating the product. This gives us $2 + \log_b b$.

Finding the Value of $\log_b b$

We can use the property of logarithms that states $\log_b b = 1$. This gives us $2 + 1$.

Simplifying the Expression

We can simplify the expression $2 + 1$ by evaluating the sum. This gives us $3$.

Conclusion

In this article, we have 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 have used the product rule, the quotient rule, and the power rule to simplify logarithmic expressions and solve logarithmic equations. We have found the value of $\log_b 4b$ to be $3$.

Final Answer

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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 some common questions and answers about 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: What are the properties of logarithms?

A: The properties of logarithms include the product rule, the quotient rule, and the power rule. The product rule states that $\log_b (xy) = \log_b x + \log_b y$. The quotient rule states that $\log_b (x/y) = \log_b x - \log_b y$. The power rule states that $\log_b x^y = y \log_b x$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to simplify the equation and isolate the variable. You can also use the inverse operation of logarithms, which is exponentiation, to solve the equation.

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

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the properties of logarithms to simplify the expression and evaluate the logarithm. You can also use a calculator to evaluate the logarithm.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is used to raise the variable to a power. For example, in the equation $\log_b x = 3$, the base is $b$.

Q: How do I change the base of a logarithm?

A: To change the base of a logarithm, you can use the change of base formula, which is $\log_b x = \frac{\log_a x}{\log_a b}$.

Q: What is the logarithm of 1?

A: The logarithm of 1 is 0, regardless of the base. This is because $b^0 = 1$ for any base $b$.

Q: What is the logarithm of 0?

A: The logarithm of 0 is undefined, regardless of the base. This is because there is no number that can be raised to a power to get 0.

Q: How do I solve a logarithmic inequality?

A: To solve a logarithmic inequality, you can use the properties of logarithms to simplify the inequality and isolate the variable. You can also use the inverse operation of logarithms, which is exponentiation, to solve the inequality.

Conclusion

In this article, we have explored some common questions and answers about logarithmic equations. We have discussed the properties of logarithms, how to solve logarithmic equations, and how to evaluate logarithmic expressions. We have also discussed the base of a logarithm, how to change the base of a logarithm, and how to solve logarithmic inequalities.

Final Answer

The final answer is $\boxed{3}$.