Select The Correct Answer.What Is The Approximate Value Of This Logarithmic Expression? Log ⁡ 8 24 \log_8 24 Lo G 8 ​ 24 A. 1.38 B. 0.90 C. 1.53 D. 0.48

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. In this article, we will delve into the world of logarithms and explore how to evaluate logarithmic expressions. We will focus on the given expression $\log_8 24$ and determine the approximate value of this expression.

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to produce a given value. For example, if we have the expression $\log_2 8$, it means that we need to find the power to which 2 must be raised to produce 8. In this case, the answer is 3, because $2^3 = 8$.

Evaluating Logarithmic Expressions

To evaluate a logarithmic expression, we need to use the definition of a logarithm. If we have the expression $\log_b x$, it means that we need to find the power to which the base $b$ must be raised to produce the value $x$. In other words, we need to solve the equation $b^y = x$ for $y$.

Using the Change of Base Formula

One of the most useful formulas in logarithms is the change of base formula. This formula allows us to change the base of a logarithmic expression from one base to another. The change of base formula is given by:

$$\log_b x = \frac{\log_a x}{\log_a b}$$

where $a$ is the new base.

Applying the Change of Base Formula

In our given expression $\log_8 24$, we can use the change of base formula to change the base from 8 to 2. This will allow us to use the logarithm of 24 with base 2, which is a common base.

$$\log_8 24 = \frac{\log_2 24}{\log_2 8}$$

Evaluating the Logarithms

Now that we have the expression in terms of base 2, we can evaluate the logarithms. We know that $\log_2 24$ is the power to which 2 must be raised to produce 24. We can find this value using a calculator or by using the fact that $2^5 = 32$ and $2^4 = 16$. Since 24 is between 16 and 32, we know that $\log_2 24$ is between 4 and 5.

Using a Calculator

To find the exact value of $\log_2 24$, we can use a calculator. The value of $\log_2 24$ is approximately 4.585.

Evaluating the Second Logarithm

Now that we have the value of $\log_2 24$, we can evaluate the second logarithm, $\log_2 8$. We know that $2^3 = 8$, so $\log_2 8$ is equal to 3.

Substituting the Values

Now that we have the values of both logarithms, we can substitute them into the expression:

$$\log_8 24 = \frac{\log_2 24}{\log_2 8} = \frac{4.585}{3}$$

Simplifying the Expression

To simplify the expression, we can divide 4.585 by 3. This gives us a value of approximately 1.528.

Conclusion

In conclusion, we have evaluated the logarithmic expression $\log_8 24$ using the change of base formula and a calculator. We found that the approximate value of this expression is 1.528.

Answer

The correct answer is:

• A. 1.38 is incorrect
• B. 0.90 is incorrect
• C. 1.53 is incorrect
• D. 0.48 is incorrect

Introduction

In our previous article, we explored the concept of logarithmic expressions and evaluated the expression $\log_8 24$. In this article, we will continue to delve into the world of logarithms and answer some frequently asked questions.

Q: What is the difference between a logarithm and an exponent?

A: A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to produce a given value. For example, if we have the expression $\log_2 8$, it means that we need to find the power to which 2 must be raised to produce 8. In this case, the answer is 3, because $2^3 = 8$.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, we need to use the definition of a logarithm. If we have the expression $\log_b x$, it means that we need to find the power to which the base $b$ must be raised to produce the value $x$. In other words, we need to solve the equation $b^y = x$ for $y$.

Q: What is the change of base formula?

A: The change of base formula is a useful formula in logarithms that allows us to change the base of a logarithmic expression from one base to another. The change of base formula is given by:

$$\log_b x = \frac{\log_a x}{\log_a b}$$

where $a$ is the new base.

Q: How do I apply the change of base formula?

A: To apply the change of base formula, we need to identify the base of the logarithmic expression and the new base that we want to use. We can then use the formula to change the base of the expression.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log_2 x$
• $\log_10 x$
• $\log_e x$

These expressions are commonly used in mathematics and are often evaluated using a calculator.

Q: How do I evaluate a logarithmic expression with a negative base?

A: To evaluate a logarithmic expression with a negative base, we need to use the fact that the logarithm of a negative number is undefined. In other words, if we have the expression $\log_{-2} 8$, it is undefined because the base is negative.

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are inverse operations. In other words, if we have the expression $\log_b x$, it means that we need to find the power to which the base $b$ must be raised to produce the value $x$. In this case, the answer is $y$, because $b^y = x$.

Q: How do I use a calculator to evaluate a logarithmic expression?

A: To use a calculator to evaluate a logarithmic expression, we need to enter the expression into the calculator and press the "log" button. The calculator will then display the value of the expression.

Conclusion

In conclusion, we have answered some frequently asked questions about logarithmic expressions. We have explored the concept of logarithms, evaluated logarithmic expressions, and discussed the change of base formula. We hope that this article has been helpful in understanding logarithmic expressions.

Additional Resources

For more information on logarithmic expressions, we recommend the following resources:

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

We hope that this article has been helpful in understanding logarithmic expressions. If you have any further questions, please don't hesitate to ask.