Evaluate Each Expression.(a) $\log _6 \frac{1}{36}=$(b) $\log _4 64=$

Introduction

Logarithmic functions are a crucial part of mathematics, and understanding how to evaluate them is essential for solving various mathematical problems. In this article, we will focus on evaluating two logarithmic expressions: $\log _6 \frac{1}{36}$ and $\log _4 64$. We will use the properties of logarithms to simplify these expressions and find their values.

Understanding Logarithms

Before we dive into evaluating the given expressions, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$, then $y = \log _a x$. The logarithm of a number $x$ with base $a$ is the exponent to which $a$ must be raised to produce $x$.

Evaluating Expression (a)

Let's start by evaluating the expression $\log _6 \frac{1}{36}$. To simplify this expression, we can use the property of logarithms that states $\log _a \frac{1}{x} = -\log _a x$. Applying this property to our expression, we get:

$$\log _6 \frac{1}{36} = -\log _6 36$$

Now, we can use the property of logarithms that states $\log _a a^x = x$ to simplify the expression further. Since $36 = 6^2$, we can rewrite the expression as:

$$-\log _6 36 = -\log _6 6^2$$

Using the property of logarithms mentioned above, we can simplify this expression to:

$$-\log _6 6^2 = -2\log _6 6$$

Since $\log _a a = 1$, we know that $\log _6 6 = 1$. Therefore, we can simplify the expression to:

$$-2\log _6 6 = -2(1) = -2$$

Evaluating Expression (b)

Now, let's evaluate the expression $\log _4 64$. To simplify this expression, we can use the property of logarithms that states $\log _a a^x = x$. Since $64 = 4^3$, we can rewrite the expression as:

$$\log _4 64 = \log _4 4^3$$

Using the property of logarithms mentioned above, we can simplify this expression to:

$$\log _4 4^3 = 3\log _4 4$$

Since $\log _a a = 1$, we know that $\log _4 4 = 1$. Therefore, we can simplify the expression to:

$$3\log _4 4 = 3(1) = 3$$

Conclusion

In this article, we evaluated two logarithmic expressions: $\log _6 \frac{1}{36}$ and $\log _4 64$. We used the properties of logarithms to simplify these expressions and find their values. The first expression simplified to $-2$, and the second expression simplified to $3$. Understanding how to evaluate logarithmic expressions is essential for solving various mathematical problems, and we hope this article has provided a clear and concise explanation of the process.

Properties of Logarithms

Here are some key properties of logarithms that we used in this article:

• $\log _a \frac{1}{x} = -\log _a x$
• $\log _a a^x = x$
• $\log _a a = 1$

Common Logarithms

Common logarithms are logarithms with a base of 10. They are denoted by $\log x$ and are used to solve problems involving exponents and powers of 10.

Natural Logarithms

Natural logarithms are logarithms with a base of $e$. They are denoted by $\ln x$ and are used to solve problems involving exponential growth and decay.

Exercises

Here are some exercises to help you practice evaluating logarithmic expressions:

1. Evaluate the expression $\log _3 \frac{1}{27}$.
2. Evaluate the expression $\log _2 16$.
3. Evaluate the expression $\log _5 25$.

Answers

Here are the answers to the exercises:

1. $\log _3 \frac{1}{27} = -3$
2. $\log _2 16 = 4$
3. $\log _5 25 = 2$

Conclusion

Introduction

In our previous article, we evaluated two logarithmic expressions: $\log _6 \frac{1}{36}$ and $\log _4 64$. We used the properties of logarithms to simplify these expressions and find their values. In this article, we will provide a Q&A section to help you understand logarithmic expressions better.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$, then $y = \log _a x$. The logarithm of a number $x$ with base $a$ is the exponent to which $a$ must be raised to produce $x$.

Q: What are the properties of logarithms?

A: There are several properties of logarithms that we used in our previous article. Here are some of the key properties:

• $\log _a \frac{1}{x} = -\log _a x$
• $\log _a a^x = x$
• $\log _a a = 1$

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to use the properties of logarithms. Here are the steps to follow:

1. Simplify the expression using the properties of logarithms.
2. Use the property $\log _a a^x = x$ to simplify the expression.
3. Use the property $\log _a \frac{1}{x} = -\log _a x$ to simplify the expression.
4. Use the property $\log _a a = 1$ to simplify the expression.

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with a base of 10. It is denoted by $\log x$ and is used to solve problems involving exponents and powers of 10. A natural logarithm is a logarithm with a base of $e$. It is denoted by $\ln x$ and is used to solve problems involving exponential growth and decay.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use the properties of logarithms. Here are the steps to follow:

1. Simplify the equation using the properties of logarithms.
2. Use the property $\log _a a^x = x$ to simplify the equation.
3. Use the property $\log _a \frac{1}{x} = -\log _a x$ to simplify the equation.
4. Use the property $\log _a a = 1$ to simplify the equation.

Q: What are some common logarithmic expressions?

A: Here are some common logarithmic expressions:

• $\log _a a^x = x$
• $\log _a \frac{1}{x} = -\log _a x$
• $\log _a a = 1$
• $\log _a b = \frac{\log c}{\log a}$

Q: How do I use a calculator to evaluate a logarithmic expression?

A: To use a calculator to evaluate a logarithmic expression, you need to follow these steps:

1. Enter the expression into the calculator.
2. Use the logarithm button to evaluate the expression.
3. Use the exponent button to evaluate the expression.

Conclusion

In this article, we provided a Q&A section to help you understand logarithmic expressions better. We covered topics such as the definition of a logarithm, the properties of logarithms, and how to evaluate a logarithmic expression. We also provided some common logarithmic expressions and explained how to use a calculator to evaluate a logarithmic expression. We hope this article has provided a clear and concise explanation of the process and has helped you understand logarithmic expressions better.

Exercises

Here are some exercises to help you practice evaluating logarithmic expressions:

1. Evaluate the expression $\log _3 \frac{1}{27}$.
2. Evaluate the expression $\log _2 16$.
3. Evaluate the expression $\log _5 25$.

Answers

Here are the answers to the exercises:

1. $\log _3 \frac{1}{27} = -3$
2. $\log _2 16 = 4$
3. $\log _5 25 = 2$

Conclusion

In this article, we provided a Q&A section to help you understand logarithmic expressions better. We covered topics such as the definition of a logarithm, the properties of logarithms, and how to evaluate a logarithmic expression. We also provided some common logarithmic expressions and explained how to use a calculator to evaluate a logarithmic expression. We hope this article has provided a clear and concise explanation of the process and has helped you understand logarithmic expressions better.