Complete The Steps To Evaluate $\log _7 98$, Given $\log _7 2 \approx 0.356$.How Can You Rewrite \$\log _7 98$[/tex\] Using The Product Property?A. $7 \log 2 + 7 \log 49$ B. $\log 7 + \log 2 + \log

Introduction

Logarithms are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will explore the steps to evaluate the logarithm $\log_7 98$, given that $\log_7 2 \approx 0.356$. We will also discuss how to rewrite $\log_7 98$ using the product property.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In other words, if $x^y = z$, then $\log_x z = y$. Logarithms have several properties, including the product property, which states that $\log_a (xy) = \log_a x + \log_a y$.

Rewriting $\log_7 98$ using the Product Property

To rewrite $\log_7 98$ using the product property, we need to express $98$ as a product of two numbers. We can write $98$ as $49 \times 2$. Therefore, we can rewrite $\log_7 98$ as:

$$\log_7 98 = \log_7 (49 \times 2)$$

Using the product property, we can rewrite this as:

$$\log_7 98 = \log_7 49 + \log_7 2$$

Evaluating $\log_7 49$

To evaluate $\log_7 49$, we can use the fact that $49 = 7^2$. Therefore, we can rewrite $\log_7 49$ as:

$$\log_7 49 = \log_7 (7^2)$$

Using the property of logarithms that states $\log_a a^x = x$, we can rewrite this as:

$$\log_7 49 = 2$$

Evaluating $\log_7 2$

We are given that $\log_7 2 \approx 0.356$. Therefore, we can substitute this value into our expression for $\log_7 98$:

$$\log_7 98 = \log_7 49 + \log_7 2 \approx 2 + 0.356$$

Simplifying the Expression

To simplify the expression, we can add the two numbers together:

$$\log_7 98 \approx 2.356$$

Conclusion

In this article, we have explored the steps to evaluate the logarithm $\log_7 98$, given that $\log_7 2 \approx 0.356$. We have also discussed how to rewrite $\log_7 98$ using the product property. By following these steps, we can simplify the expression and arrive at the final answer.

Final Answer

The final answer is $\boxed{2.356}$.

Additional Resources

For more information on logarithms and how to evaluate them, please refer to the following resources:

• Logarithm Wikipedia Page
• Logarithm Calculator
• Mathway Algebra Solver

Discussion

Introduction

Logarithms can be a challenging topic for many students and professionals. In this article, we will answer some of the most frequently asked questions about logarithms, covering topics such as the definition of a logarithm, how to evaluate logarithms, and how to apply logarithmic properties.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if $x^y = z$, then $\log_x z = y$. Logarithms have several properties, including the product property, which states that $\log_a (xy) = \log_a x + \log_a y$.

Q: How do I evaluate a logarithm?

A: To evaluate a logarithm, you need to find the exponent to which the base must be raised to obtain the given number. For example, to evaluate $\log_7 49$, you need to find the exponent to which 7 must be raised to obtain 49. Since $7^2 = 49$, we can conclude that $\log_7 49 = 2$.

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_a (xy) = \log_a x + \log_a y$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual numbers.

Q: How do I apply the product property to simplify complex logarithmic expressions?

A: To apply the product property, you need to express the given number as a product of two or more numbers. For example, to evaluate $\log_7 98$, you can express 98 as $49 \times 2$. Then, using the product property, you can rewrite $\log_7 98$ as $\log_7 49 + \log_7 2$.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to change the base of a logarithm from one base to another. The formula is:

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

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

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

A: To use the change of base formula, you need to substitute the given values into the formula. For example, to evaluate $\log_7 98$ using the change of base formula, you can substitute $a = 98$, $b = 7$, and $c = 10$ into the formula:

$$\log_7 98 = \frac{\log_{10} 98}{\log_{10} 7}$$

Q: What are some common logarithmic identities?

A: Some common logarithmic identities include:

• $\log_a a = 1$
• $\log_a 1 = 0$
• $\log_a (xy) = \log_a x + \log_a y$
• $\log_a (x/y) = \log_a x - \log_a y$
• $\log_a (x^y) = y \log_a x$

Conclusion

In this article, we have answered some of the most frequently asked questions about logarithms. We have covered topics such as the definition of a logarithm, how to evaluate logarithms, and how to apply logarithmic properties. We hope that this article has been helpful in clarifying any confusion you may have had about logarithms.

Additional Resources

For more information on logarithms and how to evaluate them, please refer to the following resources:

• Logarithm Wikipedia Page
• Logarithm Calculator
• Mathway Algebra Solver

Discussion

Do you have any questions about logarithms that we haven't covered in this article? Share your thoughts and ideas in the comments below!