Use The Change Of Base Formula To Compute $\log _{1 / 9} \frac{1}{6}$. Round Your Answer To The Nearest Thousandth.

Introduction

The change of base formula is a fundamental concept in mathematics, particularly in the field of logarithms. It allows us to express a logarithm in terms of another base, making it a powerful tool for solving various mathematical problems. In this article, we will explore the change of base formula and demonstrate its application in computing the logarithm of a fraction with a base of 1/9.

The Change of Base Formula

The change of base formula is given by:

$$\log_{b} a = \frac{\log_{c} a}{\log_{c} b}$$

where $a$ and $b$ are positive real numbers, and $c$ is a positive real number different from 1. This formula allows us to express a logarithm in terms of another base, $c$. The formula is valid for all real numbers $a$ and $b$, and it is a fundamental property of logarithms.

Applying the Change of Base Formula

To compute $\log_{1 / 9} \frac{1}{6}$, we can use the change of base formula. We will choose a convenient base, $c$, to simplify the computation. Let's choose $c = 10$, a common base for logarithms.

$$\log_{1 / 9} \frac{1}{6} = \frac{\log_{10} \frac{1}{6}}{\log_{10} \frac{1}{9}}$$

Computing the Logarithms

To compute the logarithms in the numerator and denominator, we can use the properties of logarithms. We know that $\log_{10} \frac{1}{6} = -\log_{10} 6$ and $\log_{10} \frac{1}{9} = -\log_{10} 9$.

$$\log_{1 / 9} \frac{1}{6} = \frac{-\log_{10} 6}{-\log_{10} 9}$$

Simplifying the Expression

We can simplify the expression by canceling out the negative signs in the numerator and denominator.

$$\log_{1 / 9} \frac{1}{6} = \frac{\log_{10} 6}{\log_{10} 9}$$

Using a Calculator

To compute the logarithms, we can use a calculator. We can enter the values into the calculator and compute the result.

$$\log_{10} 6 \approx 0.7782$$

$$\log_{10} 9 \approx 0.9542$$

Computing the Final Result

Now that we have the values of the logarithms, we can compute the final result.

$$\log_{1 / 9} \frac{1}{6} \approx \frac{0.7782}{0.9542} \approx 0.8165$$

Rounding the Result

We are asked to round the result to the nearest thousandth. Therefore, we round the result to 0.817.

Conclusion

In this article, we used the change of base formula to compute $\log_{1 / 9} \frac{1}{6}$. We chose a convenient base, $c = 10$, and used the properties of logarithms to simplify the expression. We then used a calculator to compute the logarithms and obtained the final result. We rounded the result to the nearest thousandth, as requested.

Final Answer

The final answer is $\boxed{0.817}$.

Additional Examples

The change of base formula can be used to compute logarithms with various bases. Here are a few additional examples:

• $\log_{2} 8 = \frac{\log_{10} 8}{\log_{10} 2} \approx \frac{0.9031}{0.3010} \approx 3.000$
• $\log_{3} 27 = \frac{\log_{10} 27}{\log_{10} 3} \approx \frac{1.4314}{0.4771} \approx 3.000$
• $\log_{4} 16 = \frac{\log_{10} 16}{\log_{10} 4} \approx \frac{1.2041}{0.6021} \approx 2.000$

These examples demonstrate the power of the change of base formula in computing logarithms with various bases.

Introduction

In our previous article, we explored the change of base formula and demonstrated its application in computing the logarithm of a fraction with a base of 1/9. In this article, we will answer some frequently asked questions about the change of base formula and provide additional examples to help solidify your understanding of this powerful tool.

Q&A

Q: What is the change of base formula?

A: The change of base formula is a fundamental concept in mathematics, particularly in the field of logarithms. It allows us to express a logarithm in terms of another base, making it a powerful tool for solving various mathematical problems.

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

A: To apply the change of base formula, you need to choose a convenient base, $c$, and use the formula:

$$\log_{b} a = \frac{\log_{c} a}{\log_{c} b}$$

where $a$ and $b$ are positive real numbers, and $c$ is a positive real number different from 1.

Q: What are some common bases for logarithms?

A: Some common bases for logarithms include 10, e, and 2. These bases are commonly used in mathematics and are often used as the base for logarithmic functions.

Q: Can I use the change of base formula to compute logarithms with any base?

A: Yes, you can use the change of base formula to compute logarithms with any base. However, you need to choose a convenient base, $c$, that is easy to work with.

Q: How do I choose a convenient base, $c$?

A: To choose a convenient base, $c$, you need to consider the properties of the base and the logarithm you are trying to compute. For example, if you are trying to compute a logarithm with a base of 2, it may be easier to choose a base of 10, since 10 is a power of 2.

Q: Can I use the change of base formula to compute logarithms with negative bases?

A: No, you cannot use the change of base formula to compute logarithms with negative bases. The change of base formula is only valid for positive real numbers.

Q: Can I use the change of base formula to compute logarithms with complex bases?

A: Yes, you can use the change of base formula to compute logarithms with complex bases. However, you need to be careful when working with complex numbers, as they can be difficult to work with.

Additional Examples

Here are a few additional examples of using the change of base formula to compute logarithms:

• $\log_{3} 27 = \frac{\log_{10} 27}{\log_{10} 3} \approx \frac{1.4314}{0.4771} \approx 3.000$
• $\log_{4} 16 = \frac{\log_{10} 16}{\log_{10} 4} \approx \frac{1.2041}{0.6021} \approx 2.000$
• $\log_{5} 25 = \frac{\log_{10} 25}{\log_{10} 5} \approx \frac{1.3979}{0.6989} \approx 2.000$

These examples demonstrate the power of the change of base formula in computing logarithms with various bases.

Conclusion

In this article, we answered some frequently asked questions about the change of base formula and provided additional examples to help solidify your understanding of this powerful tool. We hope that this article has been helpful in your studies of logarithms and the change of base formula.

Final Answer

The final answer is $\boxed{0.817}$.

Additional Resources

If you are interested in learning more about the change of base formula and logarithms, we recommend the following resources:

• Khan Academy: Logarithms
• Mathway: Change of Base Formula
• Wolfram Alpha: Change of Base Formula

These resources provide a comprehensive overview of the change of base formula and logarithms, and can be a valuable resource for students and professionals alike.