What Is The Value Of $\log_6 \frac{1}{36}$?A. -6 B. -2 C. 2 D. 6

Introduction

In mathematics, logarithms are a fundamental concept that helps us solve equations and express complex relationships in a simpler form. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In this article, we will explore the value of $\log_6 \frac{1}{36}$ and understand the underlying mathematical concepts.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In other words, if we have a number $x$ and a base $b$, then the logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$. This can be represented mathematically as:

$$\log_b x = y \iff b^y = x$$

For example, if we have $\log_2 8$, then we are looking for the exponent to which 2 must be raised to produce 8. Since $2^3 = 8$, we can conclude that $\log_2 8 = 3$.

Evaluating $\log_6 \frac{1}{36}$

To evaluate $\log_6 \frac{1}{36}$, we need to understand that $\frac{1}{36}$ can be expressed as a power of 6. Since $6^2 = 36$, we can rewrite $\frac{1}{36}$ as $\frac{1}{6^2}$. Using the properties of exponents, we can rewrite this as $6^{-2}$.

Applying the Definition of Logarithm

Now that we have expressed $\frac{1}{36}$ as a power of 6, we can apply the definition of logarithm to evaluate $\log_6 \frac{1}{36}$. Since $\frac{1}{36} = 6^{-2}$, we can conclude that:

$$\log_6 \frac{1}{36} = \log_6 6^{-2}$$

Using the property of logarithms that states $\log_b b^x = x$, we can simplify this expression to:

$$\log_6 \frac{1}{36} = -2$$

Conclusion

In conclusion, the value of $\log_6 \frac{1}{36}$ is -2. This is because $\frac{1}{36}$ can be expressed as a power of 6, specifically $6^{-2}$. By applying the definition of logarithm and using the properties of exponents, we can evaluate the logarithm and arrive at the correct answer.

Frequently Asked Questions

• What is the value of $\log_6 \frac{1}{36}$?
• How do we evaluate logarithms?
• What are the properties of logarithms?

Final Answer

The final answer is $\boxed{-2}$.

Introduction

Logarithms are a fundamental concept in mathematics that helps us solve equations and express complex relationships in a simpler form. In our previous article, we explored the value of $\log_6 \frac{1}{36}$ and understood the underlying mathematical concepts. In this article, we will address some frequently asked questions about logarithms and provide a deeper understanding of this important mathematical concept.

Q&A

Q: What is the value of $\log_6 \frac{1}{36}$?

A: The value of $\log_6 \frac{1}{36}$ is -2. This is because $\frac{1}{36}$ can be expressed as a power of 6, specifically $6^{-2}$. By applying the definition of logarithm and using the properties of exponents, we can evaluate the logarithm and arrive at the correct answer.

Q: How do we evaluate logarithms?

A: To evaluate logarithms, we need to understand that a logarithm is the inverse operation of exponentiation. In other words, if we have a number $x$ and a base $b$, then the logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$. This can be represented mathematically as:

$$\log_b x = y \iff b^y = x$$

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• $\log_b b^x = x$
• $\log_b 1 = 0$
• $\log_b b = 1$
• $\log_b \frac{1}{x} = -\log_b x$
• $\log_b (xy) = \log_b x + \log_b y$

Q: How do we simplify logarithmic expressions?

A: To simplify logarithmic expressions, we can use the properties of logarithms. For example, if we have $\log_2 (3 \cdot 4)$, we can simplify this expression by using the property $\log_b (xy) = \log_b x + \log_b y$. This gives us:

$$\log_2 (3 \cdot 4) = \log_2 3 + \log_2 4$$

Q: What is the difference between a logarithm and an exponent?

A: A logarithm and an exponent are inverse operations. In other words, if we have a number $x$ and a base $b$, then the logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$. This can be represented mathematically as:

$$\log_b x = y \iff b^y = x$$

Q: How do we use logarithms in real-world applications?

A: Logarithms are used in a wide range of real-world applications, including:

• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithms are used to calculate the power of a signal and the frequency of a wave.

Conclusion

In conclusion, logarithms are a fundamental concept in mathematics that helps us solve equations and express complex relationships in a simpler form. By understanding the properties of logarithms and how to evaluate them, we can use logarithms in a wide range of real-world applications. We hope that this article has provided a deeper understanding of logarithms and has answered some of the frequently asked questions about this important mathematical concept.

Final Answer

The final answer is $\boxed{-2}$.