Which Expression Is Equivalent To $\log_{12} \frac{x^4 \sqrt{x^3-2}}{(x+1)^5}$?A. $4 \log_{12} X + \frac{1}{2} \log_{12}(x^3-2) - 5 \log_{12}(x-1$\]B. $4 \log_{12} X + \frac{1}{2} \log_{12} \frac{x^3}{2} - 5 \log_{12} X +

Which Expression is Equivalent to $\log_{12} \frac{x^4 \sqrt{x^3-2}}{(x+1)^5}$?

In this article, we will explore the concept of logarithmic expressions and how to simplify them. We will focus on the given expression $\log_{12} \frac{x^4 \sqrt{x^3-2}}{(x+1)^5}$ and determine which of the provided options is equivalent to it.

Understanding Logarithmic Expressions

A logarithmic expression is a mathematical operation that represents the power to which a base number must be raised to produce a given value. In other words, it is the inverse operation of exponentiation. The general form of a logarithmic expression is $\log_b a = c$, where $b$ is the base, $a$ is the argument, and $c$ is the result.

Simplifying Logarithmic Expressions

To simplify a logarithmic expression, we can use the properties of logarithms. These properties include:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$

Applying the Properties of Logarithms

Let's apply the properties of logarithms to the given expression $\log_{12} \frac{x^4 \sqrt{x^3-2}}{(x+1)^5}$.

First, we can rewrite the expression as $\log_{12} \frac{x^4 (x^3-2)^{\frac{1}{2}}}{(x+1)^5}$.

Using the product property, we can rewrite the expression as $\log_{12} x^4 + \log_{12} (x^3-2)^{\frac{1}{2}} - \log_{12} (x+1)^5$.

Using the power property, we can rewrite the expression as $4 \log_{12} x + \frac{1}{2} \log_{12} (x^3-2) - 5 \log_{12} (x+1)$.

Comparing with the Options

Now, let's compare the simplified expression with the provided options.

Option A: $4 \log_{12} x + \frac{1}{2} \log_{12} (x^3-2) - 5 \log_{12} (x-1)$

Option B: $4 \log_{12} x + \frac{1}{2} \log_{12} \frac{x^3}{2} - 5 \log_{12} x$

Based on the simplified expression, we can conclude that the correct option is:

Option A: $4 \log_{12} x + \frac{1}{2} \log_{12} (x^3-2) - 5 \log_{12} (x-1)$

This option matches the simplified expression exactly, while the other options do not.

The concept of logarithmic expressions is a fundamental aspect of mathematics, and understanding how to simplify them is crucial for solving problems in various fields, including science, engineering, and economics.

In this article, we have demonstrated how to simplify a logarithmic expression using the properties of logarithms. We have also compared the simplified expression with the provided options and concluded that Option A is the correct answer.

In conclusion, logarithmic expressions are a powerful tool for solving mathematical problems. By understanding how to simplify them, we can unlock new insights and solutions to complex problems. We hope that this article has provided a clear and concise explanation of the concept and has helped readers to better understand the properties of logarithms.

• [1] "Logarithmic Expressions" by Math Open Reference
• [2] "Properties of Logarithms" by Khan Academy
• [3] "Logarithmic Identities" by Wolfram MathWorld

• [1] "Logarithmic Expressions" by Mathway
• [2] "Properties of Logarithms" by Purplemath
• [3] "Logarithmic Identities" by IXL
  Logarithmic Expressions: A Q&A Guide =====================================

In our previous article, we explored the concept of logarithmic expressions and how to simplify them. In this article, we will provide a Q&A guide to help readers better understand the properties of logarithms and how to apply them to solve problems.

Q: What is a logarithmic expression?

A: A logarithmic expression is a mathematical operation that represents the power to which a base number must be raised to produce a given value.

Q: What are the properties of logarithms?

A: The properties of logarithms include:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \frac{x}{y} = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithms. Here's a step-by-step guide:

1. Identify the base and argument of the logarithmic expression.
2. Apply the product property to simplify the expression.
3. Apply the quotient property to simplify the expression.
4. Apply the power property to simplify the expression.

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression represents the power to which a base number must be raised to produce a given value, while an exponential expression represents the result of raising a base number to a given power.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the value of the argument that corresponds to the given base. For example, if you have the expression $\log_{10} 100$, you need to find the value of $x$ such that $10^x = 100$.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log_{10} x$
• $\log_{e} x$
• $\log_{2} x$

Q: How do I use logarithmic expressions in real-world problems?

A: Logarithmic expressions are used in a variety of real-world problems, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate pH levels and concentrations of solutions.
• Engineering: Logarithmic expressions are used to calculate stress and strain on materials.

Q: What are some common mistakes to avoid when working with logarithmic expressions?

A: Some common mistakes to avoid when working with logarithmic expressions include:

• Forgetting to apply the product property
• Forgetting to apply the quotient property
• Forgetting to apply the power property

In this Q&A guide, we have provided answers to common questions about logarithmic expressions and how to apply the properties of logarithms to solve problems. We hope that this guide has been helpful in clarifying the concept and providing a better understanding of the properties of logarithms.

• [1] "Logarithmic Expressions" by Math Open Reference
• [2] "Properties of Logarithms" by Khan Academy
• [3] "Logarithmic Identities" by Wolfram MathWorld

• [1] Simplify the expression $\log_{10} \frac{x^2}{y^3}$
• [2] Evaluate the expression $\log_{e} 100$
• [3] Simplify the expression $\log_{2} (x^3 + y^3)$

• [1] $2 \log_{10} x - 3 \log_{10} y$
• [2] $4.605$
• [3] $\log_{2} (x+y) + \log_{2} (x^2 - xy + y^2)$