Use The Properties Of Logarithms To Expand The Following Expression:$\log \left(\frac{2 X^5}{(7+x)^4}\right$\]Your Answer Should Not Have Radicals Or Exponents. You May Assume That All Variables Are Positive.

Feb 28, 2025 by ADMIN 209 views

**Using Properties of Logarithms to Expand an Expression**

Introduction

Logarithms are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and economics. One of the key properties of logarithms is the ability to expand complex expressions into simpler ones. In this article, we will explore how to use the properties of logarithms to expand the given expression: $\log \left(\frac{2 x^5}{(7+x)^4}\right)$ . We will assume that all variables are positive and that the expression does not contain any radicals or exponents.

The Properties of Logarithms

Before we dive into expanding the given expression, let's review the properties of logarithms that we will use. The properties of logarithms are as follows:

Product Property: $\log (ab) = \log a + \log b$
Quotient Property: $\log \left(\frac{a}{b}\right) = \log a - \log b$
Power Property: $\log a^b = b \log a$

These properties will be the foundation of our expansion process.

Expanding the Expression

Now that we have reviewed the properties of logarithms, let's apply them to expand the given expression. We will start by using the quotient property to separate the numerator and denominator.

$\log \left(\frac{2 x^5}{(7+x)^4}\right) = \log (2 x^5) - \log ((7+x)^4)$

Next, we will use the product property to expand the first term.

$\log (2 x^5) = \log 2 + \log x^5$

Now, we will use the power property to expand the second term.

$\log ((7+x)^4) = 4 \log (7+x)$

Substituting these expansions back into the original expression, we get:

$\log \left(\frac{2 x^5}{(7+x)^4}\right) = \log 2 + \log x^5 - 4 \log (7+x)$

Simplifying the Expression

We can simplify the expression further by using the power property to expand the second term.

$\log x^5 = 5 \log x$

Substituting this expansion back into the original expression, we get:

$\log \left(\frac{2 x^5}{(7+x)^4}\right) = \log 2 + 5 \log x - 4 \log (7+x)$

Conclusion

In this article, we used the properties of logarithms to expand the given expression: $\log \left(\frac{2 x^5}{(7+x)^4}\right)$ . We started by using the quotient property to separate the numerator and denominator, and then applied the product and power properties to expand the terms. Finally, we simplified the expression by using the power property to expand the second term. The resulting expression is:

$\log \left(\frac{2 x^5}{(7+x)^4}\right) = \log 2 + 5 \log x - 4 \log (7+x)$

This expression is now in a simpler form, and it does not contain any radicals or exponents.

Example Use Cases

The expanded expression can be used in a variety of applications, including:

Scientific Calculations: The expression can be used to calculate the logarithm of a complex function in scientific calculations.
Engineering Applications: The expression can be used to model and analyze complex systems in engineering applications.
Economic Modeling: The expression can be used to model and analyze economic systems, including the behavior of stock prices and interest rates.

Conclusion

Q: What are the properties of logarithms?

A: The properties of logarithms are:

Product Property: $\log (ab) = \log a + \log b$
Quotient Property: $\log \left(\frac{a}{b}\right) = \log a - \log b$
Power Property: $\log a^b = b \log a$

Q: How do I use the properties of logarithms to expand an expression?

A: To expand an expression using the properties of logarithms, follow these steps:

Use the quotient property to separate the numerator and denominator.
Use the product property to expand the terms.
Use the power property to expand the terms.

Q: What is the difference between the product and quotient properties?

A: The product property states that $\log (ab) = \log a + \log b$ , while the quotient property states that $\log \left(\frac{a}{b}\right) = \log a - \log b$ . The product property is used to expand expressions with multiple terms, while the quotient property is used to separate the numerator and denominator.

Q: Can I use the properties of logarithms to simplify expressions with radicals or exponents?

A: No, the properties of logarithms are only applicable to expressions without radicals or exponents. If an expression contains radicals or exponents, you will need to use other mathematical techniques to simplify it.

Q: How do I apply the power property to expand an expression?

A: To apply the power property, follow these steps:

Identify the term that contains the exponent.
Use the power property to expand the term: $\log a^b = b \log a$ .

Q: Can I use the properties of logarithms to solve equations?

A: Yes, the properties of logarithms can be used to solve equations. By applying the properties of logarithms, you can simplify the equation and isolate the variable.

Q: What are some common applications of the properties of logarithms?

A: The properties of logarithms have numerous applications in various fields, including:

Scientific Calculations: The properties of logarithms are used to calculate the logarithm of complex functions in scientific calculations.
Engineering Applications: The properties of logarithms are used to model and analyze complex systems in engineering applications.
Economic Modeling: The properties of logarithms are used to model and analyze economic systems, including the behavior of stock prices and interest rates.

Q: How do I know which property to use when expanding an expression?

A: To determine which property to use, follow these steps:

Identify the type of expression you are working with (e.g., product, quotient, power).
Choose the property that corresponds to the type of expression.
Apply the property to expand the expression.

Q: Can I use the properties of logarithms to expand expressions with negative numbers?

A: Yes, the properties of logarithms can be used to expand expressions with negative numbers. However, you will need to use the properties of logarithms in conjunction with other mathematical techniques to simplify the expression.

Q: How do I simplify expressions with multiple logarithms?

A: To simplify expressions with multiple logarithms, follow these steps:

Use the product property to combine the logarithms.
Use the power property to expand the terms.
Simplify the expression using the properties of logarithms.

Q: Can I use the properties of logarithms to solve inequalities?

A: Yes, the properties of logarithms can be used to solve inequalities. By applying the properties of logarithms, you can simplify the inequality and isolate the variable.