Problem 515 - Precalculus

Question

Rewrite \(\ln \left(\frac{1}{x^{5}}\right)\) as a product.

Step-by-Step Solution

Answer

\(-5 \ln(x)\)

Step 1: Recognize the Given Expression

The expression we have is \( \ln \left(\frac{1}{x^5}\right) \). We're asked to rewrite this as a product.

Step 2: Apply the Property of Logarithms for Fractions

Recall that \( \ln \left(\frac{1}{a}\right) = -\ln(a) \). Apply this property to the expression: \( \ln \left(\frac{1}{x^5}\right) = - \ln(x^5) \).

Step 3: Apply the Power Rule of Logarithms

Use the power rule \( \ln(a^b) = b \ln(a) \). Hence, \( -\ln(x^5) \) can be expanded to: \( -5 \ln(x) \).

Step 4: Write the Expression as a Product

The expression \( -5 \ln(x) \) is already a rewritten form as a product, where \(-5\) is the coefficient multiplied by \(\ln(x)\).

Key Concepts

Logarithm Properties

Logarithms have several properties that make them incredibly useful when dealing with mathematical expressions, especially when simplifying complex forms. One essential property involves fractions in logarithmic expressions. It states that if you take the logarithm of a fraction, it can be rewritten using this simple rule:

• \( \ln \left( \frac{1}{a} \right) = -\ln(a) \)

This rule reflects how logarithms treat division as a subtraction operation, and hence a fraction's logarithm becomes the negative log of its denominator.
This property is particularly helpful in simplifying expressions such as \( \ln \left( \frac{1}{x^5} \right) \). By applying this rule, the expression becomes \( -\ln(x^5) \).
Understanding this aspect can simplify analyzing and transforming more complex logarithmic expressions.

Power Rule

The power rule is another handy property when working with logarithms, especially when dealing with powers and exponents. It states:

• \( \ln(a^b) = b \ln(a) \)

This rule explains how a logarithm applied to a power is equivalent to multiplying the exponent by the logarithm of the base.
This property is crucial when we encounter expressions like \( \ln(x^5) \). Through applying the power rule, we can transform it into \( 5 \ln(x) \). This transformation makes it easier to manipulate the expression further and is key in expressing logarithms in product form.
In our given problem, this allowed us to take the expression \( -\ln(x^5) \) and rewrite it as \( -5 \ln(x) \). This highlights the efficiency power rule brings into logarithmic transformations.

Natural Logarithm

The natural logarithm is a specific logarithm with the base \(e\), where \(e\) is approximately 2.71828. It's noted as \(\ln(x)\) and is often used in mathematical, scientific, and engineering fields.
Natural logarithms have properties similar to logarithms of other bases but are especially significant in calculus and higher mathematics because of their natural properties with exponential functions.

• The increase rate related to an exponential growth process is proportional to the amount present.
• It effectively converts multiplicative relationships into additive ones, simplifying the process of solving complex equations.

In our problem, the expression \( \ln \left( \frac{1}{x^5} \right) \) involves a natural logarithm. Utilizing the properties of natural logarithms allowed us to apply both fraction and power rules smoothly.
Mastering the use of natural logarithms helps in wide-ranging mathematical problems, particularly those involving growth processes and decay.