Write The Expression As A Single Logarithm. Express Powers As Factors.$\log_7 \sqrt{x} - \log_7 X^4$\log_7 \sqrt{x} - \log_7 X^4 = \square$(Type An Exact Answer. Use Integers Or Fractions For Any Numbers.)

Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into manageable parts. In this article, we will focus on expressing powers as factors and simplifying logarithmic expressions using the properties of logarithms.

Understanding Logarithmic Properties

Before we dive into simplifying the given expression, it's essential to understand the properties of logarithms. The two main properties we will use are:

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

Expressing Powers as Factors

The given expression is $\log_7 \sqrt{x} - \log_7 x^4$. To express powers as factors, we need to rewrite the square root as a power of $x$. We know that $\sqrt{x} = x^{\frac{1}{2}}$. Therefore, we can rewrite the expression as:

$\log_7 x^{\frac{1}{2}} - \log_7 x^4$

Simplifying the Expression

Now that we have expressed the powers as factors, we can simplify the expression using the properties of logarithms. We will use the quotient property to combine the two logarithmic terms:

$\log_7 x^{\frac{1}{2}} - \log_7 x^4 = \log_7 \left(\frac{x^{\frac{1}{2}}}{x^4}\right)$

Applying the Quotient Property

To simplify the expression inside the logarithm, we will apply the quotient property:

$\log_7 \left(\frac{x^{\frac{1}{2}}}{x^4}\right) = \log_7 x^{\frac{1}{2} - 4}$

Simplifying the Exponent

Now that we have simplified the expression inside the logarithm, we can simplify the exponent:

$\log_7 x^{\frac{1}{2} - 4} = \log_7 x^{-\frac{15}{2}}$

Conclusion

In this article, we have simplified the logarithmic expression $\log_7 \sqrt{x} - \log_7 x^4$ using the properties of logarithms. We expressed powers as factors and applied the quotient property to combine the two logarithmic terms. The final simplified expression is $\log_7 x^{-\frac{15}{2}}$. This expression can be further simplified by applying the power property of logarithms.

Final Answer

$\boxed{\log_7 x^{-\frac{15}{2}}}$

Additional Tips and Tricks

• When simplifying logarithmic expressions, it's essential to express powers as factors and apply the properties of logarithms.
• The product property and quotient property are essential tools for simplifying logarithmic expressions.
• When working with negative exponents, remember that $x^{-n} = \frac{1}{x^n}$.

Common Mistakes to Avoid

• Failing to express powers as factors can lead to incorrect simplifications.
• Not applying the properties of logarithms correctly can result in incorrect answers.
• Not simplifying the exponent can lead to complex and difficult-to-read expressions.

Conclusion

Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into manageable parts. In this article, we will answer some common questions related to logarithmic expressions and provide tips and tricks for simplifying them.

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

A: A logarithmic expression is an expression that involves the logarithm of a number, while an exponential expression is an expression that involves the exponentiation of a number. For example, $\log_7 x$ is a logarithmic expression, while $7^x$ is an exponential expression.

Q: What are the properties of logarithms?

A: The two main properties of logarithms are:

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

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you need to express powers as factors and apply the properties of logarithms. Here are the steps:

1. Express powers as factors: Rewrite the expression using the power property of logarithms.
2. Apply the product property: Combine the logarithmic terms using the product property.
3. Apply the quotient property: Combine the logarithmic terms using the quotient property.
4. Simplify the exponent: Simplify the exponent by applying the rules of exponents.

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

A: A logarithmic expression and an exponential expression with the same base are inverse functions. For example, $\log_7 x$ and $7^x$ are inverse functions. This means that if $y = \log_7 x$, then $x = 7^y$.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the value of the expression. Here are the steps:

1. Check if the expression is in logarithmic form: Make sure the expression is in the form $\log_b x$.
2. Check if the base is a positive number: Make sure the base is a positive number.
3. Check if the argument is a positive number: Make sure the argument is a positive number.
4. Evaluate the expression: Use a calculator or a logarithmic table to evaluate the expression.

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

A: Here are some common mistakes to avoid when simplifying logarithmic expressions:

• Failing to express powers as factors can lead to incorrect simplifications.
• Not applying the properties of logarithms correctly can result in incorrect answers.
• Not simplifying the exponent can lead to complex and difficult-to-read expressions.

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

A: Logarithmic expressions are used in a variety of real-world applications, 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 in materials.

Conclusion

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms and the ability to express powers as factors, we can break down these expressions into manageable parts. By following the steps outlined in this article, you can simplify complex logarithmic expressions and arrive at the correct answer. Remember to express powers as factors, apply the properties of logarithms, and simplify the exponent to arrive at the final answer.