Compute The Following Logarithm:$\[ \log_5(-125) = \\]A. \[$-5\$\] B. 5 C. 3 D. \[$-3\$\] E. There Is No Solution.

Introduction

Logarithms are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. In this article, we will delve into the world of logarithms and explore how to compute logarithms, with a focus on the logarithm $\log_5(-125)$.

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to produce a given value. For example, if we have the equation $x = 2^y$, then the logarithm of $x$ with base $2$ is $y$. This can be written as $\log_2(x) = y$.

Properties of Logarithms

Logarithms have several important properties that make them useful in various mathematical and scientific applications. Some of the key properties of logarithms include:

• The product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• The quotient rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• The power rule: $\log_b(x^y) = y\log_b(x)$

Computing Logarithms

Computing logarithms can be a straightforward process, but it requires a good understanding of the properties of logarithms. Here are the steps to compute a logarithm:

1. Check if the base and the argument are positive: Logarithms are only defined for positive real numbers. If the base or the argument is negative, then the logarithm is undefined.
2. Use the product rule: If the argument is a product of two numbers, then we can use the product rule to simplify the logarithm.
3. Use the quotient rule: If the argument is a quotient of two numbers, then we can use the quotient rule to simplify the logarithm.
4. Use the power rule: If the argument is a power of a number, then we can use the power rule to simplify the logarithm.

Computing $\log_5(-125)$

Now that we have a good understanding of logarithms and their properties, let's compute $\log_5(-125)$. To do this, we need to follow the steps outlined above.

1. Check if the base and the argument are positive: The base is $5$, which is positive, and the argument is $-125$, which is negative. Since the argument is negative, the logarithm is undefined.
2. Use the product rule: We can rewrite $-125$ as $(-1)(125)$. Using the product rule, we get $\log_5(-125) = \log_5(-1) + \log_5(125)$.
3. Use the quotient rule: We can rewrite $125$ as $\frac{125}{1}$. Using the quotient rule, we get $\log_5(125) = \log_5(125) - \log_5(1)$.
4. Use the power rule: We can rewrite $125$ as $5^3$. Using the power rule, we get $\log_5(125) = 3\log_5(5)$.

Simplifying the Expression

Now that we have simplified the expression, we can see that $\log_5(-125) = \log_5(-1) + 3\log_5(5)$. Since $\log_5(5) = 1$, we can simplify the expression further to get $\log_5(-125) = \log_5(-1) + 3$.

Conclusion

In conclusion, computing logarithms requires a good understanding of the properties of logarithms. By following the steps outlined above, we can simplify complex logarithmic expressions and arrive at a solution. In this article, we computed the logarithm $\log_5(-125)$ and arrived at the conclusion that it is undefined.

Final Answer

Introduction

Logarithms can be a complex and intimidating topic, but with the right guidance, they can be understood and applied with ease. In this article, we will answer some of the most frequently asked questions about logarithms, providing a comprehensive guide to help you master this fundamental concept in mathematics.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to produce a given value. For example, if we have the equation $x = 2^y$, then the logarithm of $x$ with base $2$ is $y$. This can be written as $\log_2(x) = y$.

Q: What are the properties of logarithms?

A: Logarithms have several important properties that make them useful in various mathematical and scientific applications. Some of the key properties of logarithms include:

• The product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• The quotient rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• The power rule: $\log_b(x^y) = y\log_b(x)$

Q: How do I compute a logarithm?

A: Computing logarithms can be a straightforward process, but it requires a good understanding of the properties of logarithms. Here are the steps to compute a logarithm:

1. Check if the base and the argument are positive: Logarithms are only defined for positive real numbers. If the base or the argument is negative, then the logarithm is undefined.
2. Use the product rule: If the argument is a product of two numbers, then we can use the product rule to simplify the logarithm.
3. Use the quotient rule: If the argument is a quotient of two numbers, then we can use the quotient rule to simplify the logarithm.
4. Use the power rule: If the argument is a power of a number, then we can use the power rule to simplify the logarithm.

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

A: A logarithm and an exponent are inverse operations. In other words, they are two sides of the same coin. If we have the equation $x = 2^y$, then the logarithm of $x$ with base $2$ is $y$. This can be written as $\log_2(x) = y$. On the other hand, if we have the equation $y = \log_2(x)$, then the exponent of $2$ to which $x$ must be raised to produce $y$ is $y$. This can be written as $x = 2^y$.

Q: Can logarithms be negative?

A: Yes, logarithms can be negative. In fact, the logarithm of a number with a base less than 1 can be negative. For example, if we have the equation $x = \log_0.5(0.5)$, then the logarithm of $x$ with base $0.5$ is $-1$. This can be written as $\log_{0.5}(0.5) = -1$.

Q: Can logarithms be complex?

A: Yes, logarithms can be complex. In fact, the logarithm of a complex number can be a complex number itself. For example, if we have the equation $x = \log_2(2i)$, then the logarithm of $x$ with base $2$ is a complex number. This can be written as $\log_2(2i) = \frac{i\pi}{2}$.

Q: What are some common logarithmic identities?

A: Some common logarithmic identities include:

• The change of base formula: $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$
• The logarithm of a product: $\log_b(xy) = \log_b(x) + \log_b(y)$
• The logarithm of a quotient: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• The logarithm of a power: $\log_b(x^y) = y\log_b(x)$

Conclusion

In conclusion, logarithms are a fundamental concept in mathematics that can be used to solve a wide range of problems. By understanding the properties of logarithms and how to compute them, you can apply logarithms to various fields, including physics, engineering, and computer science. We hope that this article has provided you with a comprehensive guide to logarithms and has helped you to better understand this complex and fascinating topic.