Problem 30 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{a}\left(\frac{x^{2}}{y z^{3}}\right) $$

Step-by-Step Solution

Answer

\(2\log_a x - \log_a y - 3\log_a z\)

Step 1: Apply the Quotient Rule

The quotient rule for logarithms states that \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \). Here, \( M = x^2 \) and \( N = yz^3 \). Applying the quotient rule gives us: \( \log_{a}(x^2) - \log_{a}(yz^3) \).

Step 2: Apply the Product Rule on the Denominator

The product rule for logarithms states that \( \log_b(MN) = \log_b M + \log_b N \). Here, \( M = y \) and \( N = z^3 \). Applying this rule gives: \( \log_{a}(yz^3) = \log_{a}y + \log_{a}z^3 \).

Step 3: Substitute Back into the Expression

Substitute the expansion from Step 2 back into the expression from Step 1. You get: \( \log_{a}(x^2) - (\log_{a}y + \log_{a}z^3) \).

Step 4: Distribute the Negative Sign

Distribute the negative sign across the terms in the parenthesis: \( \log_{a}(x^2) - \log_{a}y - \log_{a}z^3 \).

Step 5: Apply the Power Rule

The power rule for logarithms states that \( \log_b(M^n) = n\log_b M \). Apply this to \( \log_a(x^2) \) and \( \log_a(z^3) \): \( 2\log_a x - \log_a y - 3\log_a z \).

Step 6: Final Expression

Combine all the steps to write the final expanded expression: \( 2\log_a x - \log_a y - 3\log_a z \).

Key Concepts

Quotient Rule for Logarithms

When you come across the quotient of two numbers inside a logarithm, you can apply the quotient rule to simplify the expression. The quotient rule states that the logarithm of a division, \( \log_b\left(\frac{M}{N}\right) \), can be rewritten as the difference of the logarithms: \( \log_b M - \log_b N \). This rule is extremely handy in breaking down complex logarithmic expressions into more manageable pieces.
For example, if you have \( \log_a\left(\frac{x^2}{yz^3}\right) \), you can separate it into two parts: \( \log_a(x^2) \) and \( \log_a(yz^3) \).

• This allows you to see the individual contributions of numerator and denominator to the logarithmic expression.
• Make sure to pay attention to the order: numerator minus denominator.

Starting with the quotient rule is often your first step in expanding logarithmic expressions because it sets up further application of other rules.

Product Rule for Logarithms

The product rule is useful when you have a multiplication inside a logarithm. It says: the logarithm of a product, \( \log_b(MN) \), can be split up into the sum of two logarithms: \( \log_b M + \log_b N \). Use this when the expression in the log is a multiplication of terms.
For the denominator in our example, \( \log_a(yz^3) \), you apply this rule to break it down to \( \log_a y + \log_a z^3 \).

• This rule identifies both factors in the multiplication and allows them to be expanded separately.
• Perform the expansions on both factors independently.

Understanding the product rule helps simplify many expressions where terms are being multiplied inside a logarithm.

Power Rule for Logarithms

In many logarithmic expressions, you will encounter powers, which can be made simpler using the power rule. This rule states that the logarithm of a power, \( \log_b(M^n) \), is equal to multiplying the exponent by the logarithm: \( n\log_b M \). This rule works because of the properties of exponents and logs being inverse operations.
Let's apply this to \( \log_a(x^2) \) and \( \log_a(z^3) \):

• \( \log_a(x^2) \) becomes \( 2\log_a x \) because the 2 is the exponent.
• \( \log_a(z^3) \) becomes \( 3\log_a z \), using the same principle.

Applying the power rule is typically one of the last steps to take when fully expanding a logarithmic expression. It unveils the influence of exponents within a logarithmic framework, completing the possible simplifications.