Use The Change-of-base Property To Rewrite The Given Expression: $\[ \log_{15} 88.5 = \frac{\log 88.5}{\log 15} \\](Do Not Evaluate.)Evaluate The Expression:$\[ \log_{15} 88.5 \approx 1.6557 \\](Type An Integer Or A Decimal. Do Not

Introduction

Logarithms are a fundamental concept in mathematics, and the change-of-base property is a crucial tool for working with logarithms. In this article, we will explore the change-of-base property and how it can be used to rewrite a given expression involving logarithms.

What is the Change-of-Base Property?

The change-of-base property is a mathematical formula that allows us to rewrite a logarithm in terms of another base. It states that:

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

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$.

Applying the Change-of-Base Property

Let's apply the change-of-base property to the given expression:

$$\log_{15} 88.5 = \frac{\log 88.5}{\log 15}$$

In this expression, we have a logarithm with base 15, and we want to rewrite it in terms of a common logarithm (base 10). We can use the change-of-base property to do this.

Step 1: Identify the Base and Argument

In the given expression, the base is 15, and the argument is 88.5.

Step 2: Apply the Change-of-Base Property

Using the change-of-base property, we can rewrite the expression as:

$$\log_{15} 88.5 = \frac{\log 88.5}{\log 15}$$

This is the same expression we started with, but now we have explicitly applied the change-of-base property.

Evaluating the Expression

Now that we have rewritten the expression using the change-of-base property, we can evaluate it.

$$\log_{15} 88.5 \approx 1.6557$$

This is the approximate value of the expression.

Discussion

The change-of-base property is a powerful tool for working with logarithms. It allows us to rewrite a logarithm in terms of another base, which can be useful in a variety of mathematical contexts.

Example Applications

The change-of-base property has many practical applications in mathematics and science. Here are a few examples:

• Computer Science: The change-of-base property is used in computer science to convert between different number systems, such as binary and decimal.
• Engineering: The change-of-base property is used in engineering to solve problems involving logarithmic scales, such as decibel levels.
• Biology: The change-of-base property is used in biology to analyze data involving logarithmic scales, such as population growth.

Conclusion

In this article, we have explored the change-of-base property and how it can be used to rewrite a given expression involving logarithms. We have also discussed some of the practical applications of the change-of-base property in mathematics and science.

References

• Logarithm. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Logarithm
• Change-of-Base Formula. (n.d.). In Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/logarithms/change-of-base-formula.html

Further Reading

• Logarithmic Scales. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Logarithmic_scale
• Decibel Levels. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Decibel

Glossary

• Logarithm: A mathematical function that is the inverse of exponentiation.
• Base: The number that is used as the exponent in a logarithmic function.
• Argument: The number that is being logged in a logarithmic function.
• Change-of-Base Property: A mathematical formula that allows us to rewrite a logarithm in terms of another base.
  Change-of-Base Property in Logarithms: A Comprehensive Guide ===========================================================

Q&A: Frequently Asked Questions about the Change-of-Base Property

Q: What is the change-of-base property?

A: The change-of-base property is a mathematical formula that allows us to rewrite a logarithm in terms of another base. It states that:

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

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$.

Q: Why is the change-of-base property important?

A: The change-of-base property is important because it allows us to work with logarithms in different bases. This is useful in a variety of mathematical contexts, such as solving equations, graphing functions, and analyzing data.

Q: How do I apply the change-of-base property?

A: To apply the change-of-base property, you need to identify the base and argument of the logarithm you want to rewrite. Then, you can use the formula:

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

to rewrite the logarithm in terms of another base.

Q: What are some common bases used in logarithms?

A: Some common bases used in logarithms include:

• Base 10: This is the most common base used in logarithms, and is often referred to as the "common logarithm".
• Base 2: This base is used in computer science and is often referred to as the "binary logarithm".
• Base e: This base is used in mathematics and is often referred to as the "natural logarithm".

Q: How do I evaluate a logarithm using the change-of-base property?

A: To evaluate a logarithm using the change-of-base property, you need to follow these steps:

1. Identify the base and argument of the logarithm you want to evaluate.
2. Use the change-of-base property to rewrite the logarithm in terms of another base.
3. Evaluate the rewritten logarithm using the properties of logarithms.

Q: What are some common applications of the change-of-base property?

A: Some common applications of the change-of-base property include:

• Computer Science: The change-of-base property is used in computer science to convert between different number systems, such as binary and decimal.
• Engineering: The change-of-base property is used in engineering to solve problems involving logarithmic scales, such as decibel levels.
• Biology: The change-of-base property is used in biology to analyze data involving logarithmic scales, such as population growth.

Q: What are some common mistakes to avoid when using the change-of-base property?

A: Some common mistakes to avoid when using the change-of-base property include:

• Forgetting to identify the base and argument of the logarithm: Make sure to identify the base and argument of the logarithm you want to rewrite.
• Using the wrong base: Make sure to use the correct base when rewriting the logarithm.
• Not following the properties of logarithms: Make sure to follow the properties of logarithms when evaluating the rewritten logarithm.

Q: How do I practice using the change-of-base property?

A: To practice using the change-of-base property, try the following:

• Work through examples: Try working through examples of logarithms and using the change-of-base property to rewrite them.
• Practice evaluating logarithms: Try evaluating logarithms using the change-of-base property.
• Use online resources: Try using online resources, such as calculators and worksheets, to practice using the change-of-base property.

Conclusion

In this article, we have explored the change-of-base property and how it can be used to rewrite a given expression involving logarithms. We have also discussed some of the practical applications of the change-of-base property in mathematics and science. By following the steps outlined in this article, you can master the change-of-base property and become proficient in working with logarithms.