Complete The Steps To Evaluate $\log_7 98$, Given $\log_7 2 \approx 0.356$.How Can You Rewrite $\log_7 98$ Using The Product Property?A. $7 \log 2 + 7 \log 49$ B. $\log 7 + \log 2 + \log 49$ C. $\log_7

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Introduction

Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields such as science, engineering, and finance. In this article, we will explore the steps to evaluate log⁡798\log_7 98 using the product property of logarithms.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns a value that represents the power to which a base number must be raised to produce the input number. In other words, if x=log⁡byx = \log_b y, then bx=yb^x = y.

The Product Property of Logarithms

The product property of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. Mathematically, this can be expressed as:

log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y

This property can be used to rewrite a logarithmic expression in a more convenient form.

Rewriting log⁡798\log_7 98 using the Product Property

To rewrite log⁡798\log_7 98 using the product property, we can express 9898 as a product of two numbers: 22 and 4949. This can be written as:

98=2×4998 = 2 \times 49

Using the product property, we can rewrite log⁡798\log_7 98 as:

log⁡798=log⁡7(2×49)=log⁡72+log⁡749\log_7 98 = \log_7 (2 \times 49) = \log_7 2 + \log_7 49

Evaluating log⁡72\log_7 2

We are given that log⁡72≈0.356\log_7 2 \approx 0.356. This value can be used to evaluate log⁡798\log_7 98.

Evaluating log⁡749\log_7 49

To evaluate log⁡749\log_7 49, we can use the fact that 49=7249 = 7^2. This can be written as:

49=7249 = 7^2

Using the property of logarithms that states log⁡bbx=x\log_b b^x = x, we can rewrite log⁡749\log_7 49 as:

log⁡749=log⁡772=2\log_7 49 = \log_7 7^2 = 2

Combining the Results

Now that we have evaluated log⁡72\log_7 2 and log⁡749\log_7 49, we can combine the results to evaluate log⁡798\log_7 98.

log⁡798=log⁡72+log⁡749≈0.356+2=2.356\log_7 98 = \log_7 2 + \log_7 49 \approx 0.356 + 2 = 2.356

Conclusion

In this article, we have explored the steps to evaluate log⁡798\log_7 98 using the product property of logarithms. We have rewritten log⁡798\log_7 98 as the sum of two logarithmic expressions, and then evaluated each expression separately. The final result is log⁡798≈2.356\log_7 98 \approx 2.356.

Discussion

  • What are some other ways to evaluate log⁡798\log_7 98?
  • How can the product property of logarithms be used in real-world applications?
  • What are some common mistakes to avoid when working with logarithms?

References

Q: What is the product property of logarithms?

A: The product property of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. Mathematically, this can be expressed as:

log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y

Q: How can the product property of logarithms be used to evaluate log⁡798\log_7 98?

A: To evaluate log⁡798\log_7 98, we can express 9898 as a product of two numbers: 22 and 4949. This can be written as:

98=2×4998 = 2 \times 49

Using the product property, we can rewrite log⁡798\log_7 98 as:

log⁡798=log⁡7(2×49)=log⁡72+log⁡749\log_7 98 = \log_7 (2 \times 49) = \log_7 2 + \log_7 49

Q: How can we evaluate log⁡72\log_7 2?

A: We are given that log⁡72≈0.356\log_7 2 \approx 0.356. This value can be used to evaluate log⁡798\log_7 98.

Q: How can we evaluate log⁡749\log_7 49?

A: To evaluate log⁡749\log_7 49, we can use the fact that 49=7249 = 7^2. This can be written as:

49=7249 = 7^2

Using the property of logarithms that states log⁡bbx=x\log_b b^x = x, we can rewrite log⁡749\log_7 49 as:

log⁡749=log⁡772=2\log_7 49 = \log_7 7^2 = 2

Q: What is the final result of evaluating log⁡798\log_7 98?

A: Now that we have evaluated log⁡72\log_7 2 and log⁡749\log_7 49, we can combine the results to evaluate log⁡798\log_7 98.

log⁡798=log⁡72+log⁡749≈0.356+2=2.356\log_7 98 = \log_7 2 + \log_7 49 \approx 0.356 + 2 = 2.356

Q: What are some common mistakes to avoid when working with logarithms?

A: Some common mistakes to avoid when working with logarithms include:

  • Forgetting to change the base of the logarithm when switching from one base to another.
  • Not using the correct property of logarithms to evaluate an expression.
  • Not checking the domain of the logarithm function to ensure that the input is positive.

Q: How can the product property of logarithms be used in real-world applications?

A: The product property of logarithms can be used in a variety of real-world applications, including:

  • Calculating the sound level of a loudspeaker in decibels.
  • Determining the pH of a solution in chemistry.
  • Evaluating the growth rate of a population in biology.

Q: What are some other ways to evaluate log⁡798\log_7 98?

A: Some other ways to evaluate log⁡798\log_7 98 include:

  • Using the change of base formula to change the base of the logarithm to a more convenient base.
  • Using the property of logarithms that states log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b} to change the base of the logarithm.
  • Using a calculator to evaluate the logarithm directly.

Conclusion

In this article, we have explored the steps to evaluate log⁡798\log_7 98 using the product property of logarithms. We have rewritten log⁡798\log_7 98 as the sum of two logarithmic expressions, and then evaluated each expression separately. The final result is log⁡798≈2.356\log_7 98 \approx 2.356. We have also discussed some common mistakes to avoid when working with logarithms and provided some examples of real-world applications of the product property of logarithms.