Base Formula Worksheet - 0224257) Express $\log _{\frac{1}{2}} X$ Using The Change Of Base Formula.$\log _b A=\frac{\log _c A}{\log _c B}$ Where \$b \neq 1, C \neq 1$[/tex\]

Introduction

In mathematics, the change of base formula is a fundamental concept used to express logarithms in different bases. This formula allows us to convert a logarithm from one base to another, making it a powerful tool for solving various mathematical problems. In this worksheet, we will explore the change of base formula and learn how to express a logarithm in a different base using this formula.

The Change of Base Formula

The change of base formula is given by:

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

where $b \neq 1, c \neq 1$

This formula states that the logarithm of a number $a$ with base $b$ is equal to the logarithm of $a$ with base $c$ divided by the logarithm of $b$ with base $c$. This formula can be used to express a logarithm in a different base, making it a versatile tool for solving mathematical problems.

Expressing Logarithms Using the Change of Base Formula

To express a logarithm using the change of base formula, we need to follow these steps:

1. Choose a new base $c$ that is different from the original base $b$.
2. Calculate the logarithm of $a$ with base $c$.
3. Calculate the logarithm of $b$ with base $c$.
4. Divide the logarithm of $a$ with base $c$ by the logarithm of $b$ with base $c$.

Example 1: Expressing Logarithm with Base 2

Let's say we want to express the logarithm $\log_{\frac{1}{2}} x$ using the change of base formula. We can choose a new base $c = 10$.

$$\log_{\frac{1}{2}} x = \frac{\log_{10} x}{\log_{10} \frac{1}{2}}$$

To calculate the logarithm of $\frac{1}{2}$ with base $10$, we can use the fact that $\log_{10} \frac{1}{2} = -\log_{10} 2$.

$$\log_{\frac{1}{2}} x = \frac{\log_{10} x}{-\log_{10} 2}$$

Example 2: Expressing Logarithm with Base 3

Let's say we want to express the logarithm $\log_{3} x$ using the change of base formula. We can choose a new base $c = 10$.

$$\log_{3} x = \frac{\log_{10} x}{\log_{10} 3}$$

To calculate the logarithm of $3$ with base $10$, we can use the fact that $\log_{10} 3 \approx 0.477$.

$$\log_{3} x = \frac{\log_{10} x}{0.477}$$

Conclusion

In this worksheet, we learned how to express logarithms using the change of base formula. We saw how to choose a new base, calculate the logarithm of the original base with the new base, and then divide the logarithm of the original number by the logarithm of the original base. This formula is a powerful tool for solving mathematical problems and can be used to express logarithms in different bases.

Practice Problems

1. Express the logarithm $\log_{4} x$ using the change of base formula.
2. Express the logarithm $\log_{5} x$ using the change of base formula.
3. Express the logarithm $\log_{\frac{1}{3}} x$ using the change of base formula.

Answer Key

1. $\log_{4} x = \frac{\log_{10} x}{\log_{10} 4}$
2. $\log_{5} x = \frac{\log_{10} x}{\log_{10} 5}$
3. $\log_{\frac{1}{3}} x = \frac{\log_{10} x}{-\log_{10} 3}$
   Q&A: Base Formula Worksheet =============================

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to express a logarithm in a different base. It is given by:

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

where $b \neq 1, c \neq 1$

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

A: To use the change of base formula, you need to follow these steps:

1. Choose a new base $c$ that is different from the original base $b$.
2. Calculate the logarithm of $a$ with base $c$.
3. Calculate the logarithm of $b$ with base $c$.
4. Divide the logarithm of $a$ with base $c$ by the logarithm of $b$ with base $c$.

Q: Can I use any base for the change of base formula?

A: No, you cannot use any base for the change of base formula. The base $c$ must be different from the original base $b$, and it must be a valid base for the logarithm.

Q: What are some common bases used for the change of base formula?

A: Some common bases used for the change of base formula are:

• Base 10 (decimal)
• Base 2 (binary)
• Base 8 (octal)
• Base 16 (hexadecimal)

Q: How do I choose a new base for the change of base formula?

A: To choose a new base for the change of base formula, you need to consider the following factors:

• The original base $b$ must be different from the new base $c$.
• The new base $c$ must be a valid base for the logarithm.
• The new base $c$ should be easy to work with and have a simple logarithm.

Q: Can I use the change of base formula to express a logarithm with a negative base?

A: No, you cannot use the change of base formula to express a logarithm with a negative base. The base of a logarithm must be a positive number.

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

A: Some common applications of the change of base formula are:

• Converting between different bases for logarithms
• Simplifying complex logarithmic expressions
• Solving mathematical problems involving logarithms

Q: Can I use the change of base formula to express a logarithm with a fractional base?

A: Yes, you can use the change of base formula to express a logarithm with a fractional base. However, you need to be careful when working with fractional bases, as they can lead to complex logarithmic expressions.

Q: How do I simplify a logarithmic expression using the change of base formula?

A: To simplify a logarithmic expression using the change of base formula, you need to follow these steps:

1. Choose a new base $c$ that is different from the original base $b$.
2. Calculate the logarithm of $a$ with base $c$.
3. Calculate the logarithm of $b$ with base $c$.
4. Divide the logarithm of $a$ with base $c$ by the logarithm of $b$ with base $c$.
5. Simplify the resulting expression.

Conclusion

In this Q&A article, we covered some common questions and answers related to the base formula worksheet. We discussed the change of base formula, how to use it, and some common applications of the formula. We also covered some common mistakes to avoid when using the change of base formula.