What Is The Value Of $\log_4 \frac{1}{16}$? Answer Attempt 1 Out Of 3: □ \square □

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Introduction

In mathematics, logarithms are a fundamental concept that helps us solve equations and understand the properties of numbers. The logarithm of a number is the power to which a base number must be raised to produce that number. In this article, we will explore the value of $\log_4 \frac{1}{16}$ and provide a step-by-step solution to this problem.

Understanding Logarithms

Before we dive into the solution, let's briefly review the concept of logarithms. The logarithm of a number x with base b is denoted as $\log_b x$ and is defined as the power to which b must be raised to produce x. In other words, if $b^y = x$, then $\log_b x = y$. For example, $\log_2 8 = 3$ because $2^3 = 8$.

Solution Attempt 1

To solve the problem $\log_4 \frac{1}{16}$, we can start by rewriting the expression in a more familiar form. We know that $\frac{1}{16} = 2^{-4}$, so we can rewrite the expression as $\log_4 2^{-4}$.

Using Properties of Logarithms

Now, we can use the property of logarithms that states $\log_b (x^y) = y \log_b x$. In this case, we have $\log_4 2^{-4} = -4 \log_4 2$.

Evaluating the Logarithm

To evaluate $\log_4 2$, we can use the fact that $4^{\frac{1}{2}} = 2$. This means that $\log_4 2 = \frac{1}{2}$.

Substituting the Value

Now, we can substitute the value of $\log_4 2$ into the expression $-4 \log_4 2$ to get $-4 \times \frac{1}{2} = -2$.

Conclusion

Therefore, the value of $\log_4 \frac{1}{16}$ is $-2$.

Discussion

The solution to this problem involves using the properties of logarithms to rewrite the expression in a more familiar form and then evaluating the logarithm. This problem requires a good understanding of logarithmic functions and their properties.

Related Problems

If you are interested in exploring more problems related to logarithms, here are a few examples:

  • log⁡39=?\log_3 9 = ?

  • log⁡24=?\log_2 4 = ?

  • log⁡525=?\log_5 25 = ?

Final Answer

The final answer to the problem $\log_4 \frac{1}{16}$ is $-2$.

Introduction

In our previous article, we explored the value of $\log_4 \frac{1}{16}$ and provided a step-by-step solution to this problem. In this article, we will answer some frequently asked questions about logarithms and their applications.

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

A: A logarithm is the power to which a base number must be raised to produce a given number, while an exponent is the power to which a base number is raised to produce a given number. For example, $\log_2 8 = 3$ because $2^3 = 8$.

Q: How do I evaluate a logarithm?

A: To evaluate a logarithm, you can use the properties of logarithms, such as the product rule and the quotient rule. You can also use a calculator or a logarithmic table to find the value of a logarithm.

Q: What is the base of a logarithm?

A: The base of a logarithm is the number that is raised to a power to produce a given number. For example, in the expression $\log_2 8$, the base is 2.

Q: Can I use a calculator to evaluate a logarithm?

A: Yes, you can use a calculator to evaluate a logarithm. Most calculators have a logarithm button that allows you to enter the base and the number, and then it will display the value of the logarithm.

Q: What is the logarithmic scale?

A: The logarithmic scale is a scale that uses logarithms to measure the magnitude of a quantity. It is often used in science and engineering to measure quantities such as sound levels, light levels, and earthquake magnitudes.

Q: How do I convert a logarithmic scale to a linear scale?

A: To convert a logarithmic scale to a linear scale, you can use the formula $y = \log_b x$, where y is the logarithmic value and x is the linear value. You can also use a calculator or a logarithmic table to convert a logarithmic scale to a linear scale.

Q: What are some real-world applications of logarithms?

A: Logarithms have many real-world applications, including:

  • Sound levels: Logarithms are used to measure sound levels in decibels (dB).
  • Light levels: Logarithms are used to measure light levels in lux (lx).
  • Earthquake magnitudes: Logarithms are used to measure earthquake magnitudes on the Richter scale.
  • Finance: Logarithms are used to calculate interest rates and investment returns.
  • Science: Logarithms are used to measure quantities such as pH levels, concentrations, and temperatures.

Q: Can I use logarithms to solve equations?

A: Yes, you can use logarithms to solve equations. Logarithms can be used to simplify equations and solve for unknown variables.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

  • \log_b x$: This is the logarithm of x with base b.

  • \log_b (x^y)$: This is the logarithm of x^y with base b.

  • \log_b \frac{x}{y}$: This is the logarithm of x/y with base b.

Q: Can I use logarithms to graph functions?

A: Yes, you can use logarithms to graph functions. Logarithmic scales can be used to graph functions that have a large range of values.

Conclusion

In this article, we have answered some frequently asked questions about logarithms and their applications. Logarithms are a powerful tool that can be used to solve equations, graph functions, and measure quantities. They have many real-world applications, including sound levels, light levels, earthquake magnitudes, finance, and science.

Final Answer

The final answer to the question "What is the value of $\log_4 \frac{1}{16}$?" is $-2$.