Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\log \left(\frac{x^{15} y^{13}}{z^{19}}\right)\)

Step-by-Step Solution

Answer

The expression expands to \( 15 \cdot \log(x) + 13 \cdot \log(y) - 19 \cdot \log(z) \).

Step 1: Apply the Quotient Rule for Logarithms

The Quotient Rule states that \( \log \left(\frac{a}{b}\right) = \log(a) - \log(b) \). Apply this to the given expression: \( \log \left( \frac{x^{15} y^{13}}{z^{19}} \right) = \log(x^{15} y^{13}) - \log(z^{19}) \).

Step 2: Apply the Product Rule for Logarithms

The Product Rule states that \( \log(ab) = \log(a) + \log(b) \). Apply this to \( \log(x^{15} y^{13}) \): \( \log(x^{15} y^{13}) = \log(x^{15}) + \log(y^{13}) \).

Step 3: Apply the Power Rule for Logarithms

The Power Rule states that \( \log(a^n) = n \cdot \log(a) \). Apply this to each logarithmic term: \( \log(x^{15}) = 15 \cdot \log(x) \), \( \log(y^{13}) = 13 \cdot \log(y) \), and \( \log(z^{19}) = 19 \cdot \log(z) \).

Step 4: Combine All Terms

Substitute the expanded terms into the expression: \( 15 \cdot \log(x) + 13 \cdot \log(y) - 19 \cdot \log(z) \).

Key Concepts

Quotient Rule for Logarithms

The quotient rule is a key concept in logarithms that simplifies the expression of the logarithm of a quotient. It states that the logarithm of a division can be transformed into a subtraction:

• When you have a logarithm like \( \log \left( \frac{a}{b} \right) \), split it into \( \log(a) - \log(b) \).
• This rule is based on the idea that dividing numbers corresponds to subtracting their logarithms.

Using the quotient rule helps manage complex logarithmic expressions.
For example, in our exercise, we applied this rule to split \( \log \left( \frac{x^{15} y^{13}}{z^{19}} \right) \) into two parts: \( \log(x^{15}y^{13}) - \log(z^{19}) \).
This simplified step lays the foundation for further expansion using other logarithmic rules.

Product Rule for Logarithms

The product rule assists in breaking down the logarithm of a product into a sum of logarithms. It states:

• Given a product \( ab \), the logarithm can be written as \( \log(ab) = \log(a) + \log(b) \).
• This rule is grounded in the property that multiplying numbers corresponds to adding their logarithms.

In our exercise, after using the quotient rule, we applied the product rule to the term \( \log(x^{15} y^{13}) \), expanding it to \( \log(x^{15}) + \log(y^{13}) \).
By using the product rule, each factor within the logarithm can be addressed separately, making the expression simpler to manage.

Power Rule for Logarithms

The power rule is an efficient way to handle logarithms involving exponents. According to this rule:

• The logarithm of a power \( a^n \) becomes \( n \times \log(a) \).
• This transformation is possible because exponents can be expressed as multiplicative constants outside the logarithm.

For our exercise, utilizing the power rule allowed us to further expand each term:
- \( \log(x^{15}) \) became \( 15 \cdot \log(x) \)
- \( \log(y^{13}) \) transformed into \( 13 \cdot \log(y) \)
- \( \log(z^{19}) \) changed to \( 19 \cdot \log(z) \)
Implementing this rule completes the expansion and gives a fully simplified expression.