Select The Correct Answer.What Is The Approximate Value Of This Logarithmic Expression? Log ⁡ 8 24 \log _8 24 Lo G 8 ​ 24 A. 0.90 B. 1.53 C. 0.48 D. 1.38

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Logarithmic expressions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we will delve into the world of logarithms and explore the concept of logarithmic expressions. We will also discuss how to evaluate logarithmic expressions and provide a step-by-step guide on how to solve the given problem.

What is a Logarithmic Expression?

A logarithmic expression is a mathematical expression that represents the power to which a base number must be raised to obtain a given value. In other words, it is the inverse operation of exponentiation. The logarithmic expression is denoted by the symbol "log" and is usually written as:

log_b(x) = y

where b is the base, x is the value, and y is the logarithm.

Evaluating Logarithmic Expressions

To evaluate a logarithmic expression, we need to find the value of the logarithm. This can be done using various methods, including:

  • Change of Base Formula: This formula allows us to change the base of a logarithmic expression to a more convenient base. The change of base formula is given by:

log_b(x) = (log_c(x)) / (log_c(b))

where c is the new base.

  • Logarithmic Tables: These tables provide the values of logarithms for various bases and values. We can use these tables to find the value of a logarithmic expression.

  • Calculator: Many calculators have a built-in logarithmic function that allows us to evaluate logarithmic expressions.

Solving the Given Problem

The given problem is to evaluate the logarithmic expression log⁡824\log _8 24. To solve this problem, we can use the change of base formula. We will change the base of the logarithmic expression to a more convenient base, such as 10.

Step 1: Change the Base

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

log⁡824=(log1024)/(log108)\log _8 24 = (log_{10} 24) / (log_{10} 8)

Step 2: Evaluate the Logarithms

We can use a calculator to evaluate the logarithms. The value of log⁡1024\log_{10} 24 is approximately 1.38, and the value of log⁡108\log_{10} 8 is approximately 0.90.

Step 3: Simplify the Expression

Now that we have evaluated the logarithms, we can simplify the expression by dividing the two values:

log⁡824=(1.38)/(0.90)=1.53\log _8 24 = (1.38) / (0.90) = 1.53

Conclusion

In this article, we have discussed the concept of logarithmic expressions and provided a step-by-step guide on how to evaluate the given problem. We have used the change of base formula to change the base of the logarithmic expression to a more convenient base, and then evaluated the logarithms using a calculator. The final answer is 1.53\boxed{1.53}.

Discussion

The given problem is a classic example of a logarithmic expression, and it requires the use of the change of base formula to evaluate. The change of base formula is a powerful tool that allows us to change the base of a logarithmic expression to a more convenient base. This formula is widely used in mathematics and is an essential tool for evaluating logarithmic expressions.

Key Takeaways

  • Logarithmic expressions are a fundamental concept in mathematics.
  • The change of base formula is a powerful tool for evaluating logarithmic expressions.
  • We can use a calculator to evaluate logarithms.
  • The final answer is 1.53\boxed{1.53}.

References

Additional Resources

In this article, we will provide answers to some of the most frequently asked questions about logarithmic expressions. Whether you are a student, a teacher, or simply someone who wants to learn more about logarithms, this article is for you.

Q: What is a logarithmic expression?

A: A logarithmic expression is a mathematical expression that represents the power to which a base number must be raised to obtain a given value. In other words, it is the inverse operation of exponentiation.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the change of base formula, logarithmic tables, or a calculator. The change of base formula is given by:

log_b(x) = (log_c(x)) / (log_c(b))

where c is the new base.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows you to change the base of a logarithmic expression to a more convenient base. This formula is given by:

log_b(x) = (log_c(x)) / (log_c(b))

where c is the new base.

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

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

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is the inverse operation of an exponential expression. In other words, if y = b^x, then x = log_b(y).

Q: Can I use logarithmic expressions to solve equations?

A: Yes, you can use logarithmic expressions to solve equations. For example, if you have an equation of the form:

b^x = y

you can take the logarithm of both sides to get:

x = log_b(y)

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

  • log_b(a)
  • log_b(a^c)
  • log_b(b^c)
  • log_b(a/b)

Q: Can I use logarithmic expressions to solve inequalities?

A: Yes, you can use logarithmic expressions to solve inequalities. For example, if you have an inequality of the form:

b^x > y

you can take the logarithm of both sides to get:

x > log_b(y)

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

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

  • Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
  • Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
  • Engineering: Logarithmic expressions are used to calculate the power of a signal and the frequency of a wave.

Conclusion

In this article, we have provided answers to some of the most frequently asked questions about logarithmic expressions. Whether you are a student, a teacher, or simply someone who wants to learn more about logarithms, this article is for you. We hope that you have found this article helpful and informative.

Additional Resources