Evaluate { \log \frac{1}{100}$}$.Write Your Answer As A Whole Number, Proper Fraction, Or Improper Fraction In Simplest Form.

Introduction

In this article, we will delve into the world of logarithms and evaluate the expression $\log \frac{1}{100}$. Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. The logarithm of a number is the power to which another fixed number, the base, must be raised to produce that number. In this case, we are dealing with the logarithm of $\frac{1}{100}$.

Understanding Logarithms

Before we proceed with the evaluation of $\log \frac{1}{100}$, let's briefly review the concept of logarithms. The logarithm of a number $x$ with base $b$ is denoted by $\log_b x$. It is the power to which $b$ must be raised to produce $x$. In other words, if $y = \log_b x$, then $b^y = x$. For example, if we want to find the logarithm of $100$ with base $10$, we are looking for the power to which $10$ must be raised to produce $100$. Since $10^2 = 100$, we have $\log_{10} 100 = 2$.

Evaluating $\log \frac{1}{100}$

Now that we have a good understanding of logarithms, let's focus on evaluating $\log \frac{1}{100}$. We can start by rewriting $\frac{1}{100}$ as $10^{-2}$. This is because $10^{-2}$ is equal to $\frac{1}{10^2}$, which is equal to $\frac{1}{100}$. Now, we can use the property of logarithms that states $\log_b (x^y) = y \log_b x$. In this case, we have $\log \frac{1}{100} = \log 10^{-2}$.

Using Properties of Logarithms

We can further simplify the expression $\log 10^{-2}$ by using the property of logarithms that states $\log_b (b^x) = x$. In this case, we have $\log 10^{-2} = -2 \log 10$. Since $\log 10 = 1$, we have $-2 \log 10 = -2$.

Conclusion

In conclusion, we have evaluated the expression $\log \frac{1}{100}$ and found that it is equal to $-2$. This is a whole number, and it is the simplest form of the expression.

Final Answer

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

Additional Examples

Here are a few additional examples of evaluating logarithmic expressions:

• $\log \frac{1}{1000} = \log 10^{-3} = -3$
• $\log \frac{1}{10000} = \log 10^{-4} = -4$
• $\log \frac{1}{100000} = \log 10^{-5} = -5$

These examples demonstrate the same property of logarithms that we used to evaluate $\log \frac{1}{100}$.

Applications of Logarithms

Logarithms have numerous applications in various fields such as physics, engineering, and computer science. Some of the key applications of logarithms include:

• Sound levels: Logarithms are used to measure sound levels in decibels (dB). A sound level of 0 dB is equivalent to the threshold of human hearing, while a sound level of 120 dB is equivalent to the threshold of pain.
• Seismology: Logarithms are used to measure the magnitude of earthquakes. The Richter scale is a logarithmic scale that measures the magnitude of earthquakes.
• Finance: Logarithms are used to calculate the return on investment (ROI) of a stock or a bond. The ROI is calculated as the logarithm of the ratio of the current price to the initial price.

Conclusion

In conclusion, we have evaluated the expression $\log \frac{1}{100}$ and found that it is equal to $-2$. We have also discussed the properties of logarithms and their applications in various fields. Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science.

Introduction

In this article, we will answer some frequently asked questions about logarithms. Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. We will cover a range of topics, from the basics of logarithms to their applications in real-world scenarios.

Q1: What is a logarithm?

A1: A logarithm is the power to which a base number must be raised to produce a given number. In other words, if $y = \log_b x$, then $b^y = x$. For example, if we want to find the logarithm of $100$ with base $10$, we are looking for the power to which $10$ must be raised to produce $100$. Since $10^2 = 100$, we have $\log_{10} 100 = 2$.

Q2: What is the difference between a logarithm and an exponent?

A2: A logarithm and an exponent are related but distinct concepts. An exponent is a number that is raised to a power, while a logarithm is the power to which a base number must be raised to produce a given number. For example, $10^2$ is an exponent, while $\log_{10} 100$ is a logarithm.

Q3: What are the properties of logarithms?

A3: Logarithms have several important properties, including:

• Product rule: $\log_b (xy) = \log_b x + \log_b y$
• Quotient rule: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• Power rule: $\log_b (x^y) = y \log_b x$
• Change of base rule: $\log_b x = \frac{\log_a x}{\log_a b}$

Q4: How do logarithms relate to exponents?

A4: Logarithms and exponents are closely related. In fact, the logarithm of a number is the inverse of the exponentiation operation. For example, if $y = \log_b x$, then $b^y = x$. This means that logarithms and exponents are inverse operations.

Q5: What are some common logarithmic bases?

A5: There are several common logarithmic bases, including:

• Base 10: This is the most common logarithmic base, and it is used in many mathematical and scientific applications.
• Base e: This is a natural logarithmic base, and it is used in many mathematical and scientific applications.
• Base 2: This is a binary logarithmic base, and it is used in many computer science and engineering applications.

Q6: How do logarithms apply to real-world scenarios?

A6: Logarithms have numerous applications in real-world scenarios, including:

• Sound levels: Logarithms are used to measure sound levels in decibels (dB). A sound level of 0 dB is equivalent to the threshold of human hearing, while a sound level of 120 dB is equivalent to the threshold of pain.
• Seismology: Logarithms are used to measure the magnitude of earthquakes. The Richter scale is a logarithmic scale that measures the magnitude of earthquakes.
• Finance: Logarithms are used to calculate the return on investment (ROI) of a stock or a bond. The ROI is calculated as the logarithm of the ratio of the current price to the initial price.

Q7: What are some common logarithmic functions?

A7: There are several common logarithmic functions, including:

• Natural logarithm: This is the logarithm of a number with base e. It is denoted by $\ln x$.
• Common logarithm: This is the logarithm of a number with base 10. It is denoted by $\log x$.
• Binary logarithm: This is the logarithm of a number with base 2. It is denoted by $\log_2 x$.

Q8: How do logarithms relate to other mathematical concepts?

A8: Logarithms are closely related to other mathematical concepts, including:

• Exponents: Logarithms and exponents are inverse operations.
• Trigonometry: Logarithms are used in trigonometric functions such as the logarithmic sine and cosine.
• Calculus: Logarithms are used in calculus to solve equations and optimize functions.

Conclusion

In conclusion, we have answered some frequently asked questions about logarithms. Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and computer science. We hope that this article has provided a helpful overview of logarithms and their applications.