Problem 42 - Precalculus with Limits

Question

Expanding a Logarithmic Expression In Exercises \(37-58\) , use properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{6} \frac{1}{z^{3}}$$

Step-by-Step Solution

Answer

\(-3 \cdot \log_{6}z\)

Step 1: Identify the applicable properties of logarithms

Two properties of logarithms will be useful in solving this exercise: the quotient rule and the power rule. The quotient rule states that \( \log_b(a/c) = \log_b a - \log_b c \). The power rule states that \( \log_b(a^n) = n \cdot \log_b a \).

Step 2: Apply the quotient rule

Applying the quotient rule to the given expression \( \log _{6} \frac{1}{z^{3}} \), the numerator of the fraction inside the log is 1, and the denominator is \(z^{3}\). According to the quotient rule, the logarithmic expression can be rewritten as \( \log_6 1 - \log_6 z^{3} \).

Step 3: Apply the power rule and simplify

Next, apply the power rule to \(\log_{6}z^{3}\). This then becomes \(3 \cdot \log_{6}z\), and \( \log_6 1 \) simplifies to 0 (since any number raised to the power of 0 is 1). Therefore, the final expression becomes \(0 - 3 \cdot \log_{6}z\) or simply \( -3 \cdot \log_{6}z \).

Key Concepts

Properties of Logarithms

Logarithms have several key properties that make them a powerful tool for simplifying expressions. When dealing with logarithms, understanding these properties can help you manipulate and expand complex logarithmic expressions. At the core, there are three basic properties:

• Product Rule: This states that the log of a product is the sum of the logs of the factors: \(\log_b(mn) = \log_b m + \log_b n\).
• Quotient Rule: This states that the log of a quotient is the difference of the logs: \(\log_b(a/c) = \log_b a - \log_b c\).
• Power Rule: This states that the log of a power is the exponent times the log of the base: \(\log_b(a^n) = n \cdot \log_b a\).

Recognizing when to apply each of these properties is crucial when it comes to expanding logarithmic expressions. They allow you to break down complex expressions into more manageable parts.

Quotient Rule

The quotient rule in logarithms is particularly useful when dealing with division inside a logarithmic expression. The rule states that the logarithm of a quotient \(\frac{a}{c}\) is equal to the difference between the logarithm of the numerator and the logarithm of the denominator. Mathematically, this is represented as:\[\log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c\]In the context of our exercise, this rule is applied to the expression \(\log_6 \frac{1}{z^3}\). Here, the numerator is 1, and the denominator is \(z^3\). Applying the quotient rule, the expression simplifies to \(\log_6 1 - \log_6 z^3\).

• The term \(\log_6 1\) simplifies to 0, because any number raised to the power of zero equals one.
• The remaining term is \(-\log_6 z^3\), which requires further simplification using other properties.

This rule is essential for transforming division inside a log into simpler operations, paving the way for further manipulation.

Power Rule

The power rule is invaluable when dealing with logarithms containing exponential terms. According to the power rule, when you have the logarithm of a power, you can move the exponent in front of the log as a multiplier. This property is given as:\[\log_b(a^n) = n \cdot \log_b a\]In our example, we see this with the term \(\log_6 z^3\). Using the power rule, we can transform this term to \(3 \cdot \log_6 z\). This means that the exponent 3 moves in front, simplifying the expression. The power rule thus allows you to handle complex exponents within logarithms effectively, making the expressions less cumbersome and easier to manipulate further.

• Apply this rule to move exponents to the front, turning multiplications into additions.
• It can significantly simplify expressions, especially those involving powers and roots.

The power rule turns challenging exponential terms into linear terms within logs, streamlining the solving process.