Problem 30 - Precalculus Mathematics for Calculus

Question

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 first rule to apply in expanding logarithmic expressions is the quotient rule. The quotient rule for logarithms states: \( \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \). Apply this rule to our expression: \( \log_a\left(\frac{x^2}{yz^3}\right) \). It becomes \( \log_a(x^2) - \log_a(yz^3) \).

Step 2: Apply the Product Rule

The next rule we use is the product rule. The product rule states: \( \log_b(MN) = \log_b(M) + \log_b(N) \). Apply this rule to \( \log_a(yz^3) \) to get \( \log_a(y) + \log_a(z^3) \).

Step 3: Substitute the Product Rule Expansion

Replace \( \log_a(yz^3) \) in our expression with its expanded form: \( \log_a(x^2) - (\log_a(y) + \log_a(z^3)) \). Simplifying this gives \( \log_a(x^2) - \log_a(y) - \log_a(z^3) \).

Step 4: Apply the Power Rule

The final rule we apply is the power rule, which states: \( \log_b(M^n) = n\log_b(M) \). Apply this to \( \log_a(x^2) \) and \( \log_a(z^3) \) to yield \( 2\log_a(x) \) and \( 3\log_a(z) \) respectively.

Step 5: Final Expansion

Replace the terms in the expression with the forms obtained from applying the power rule: \( 2\log_a(x) - \log_a(y) - 3\log_a(z) \). This is the expanded form of the given logarithmic expression.

Key Concepts

Laws of Logarithms

Logarithms come with some specific rules that help simplify expressions. These rules make it easier to work with complex logarithmic equations. When we talk about the Laws of Logarithms, we mainly refer to three core principles: the quotient rule, product rule, and power rule. Understanding these principles is crucial to manipulating and expanding logarithmic expressions effectively.

Each of these rules provides a method to break down a larger, complicated logarithmic expression into smaller, simpler parts. This can make calculations more manageable and can also help in identifying patterns and solutions in algebraic equations. Familiarity with these laws ensures that solving logarithmic problems becomes a more straightforward task.

Quotient Rule

The quotient rule is one of the foundational laws of logarithms. This rule is used when dealing with the logarithm of a division between two numbers or expressions. It states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator.

For any logarithm to the base \( b \), the quotient rule is written as:

• \( \log_b\left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \)

In practice, this allows us to separate the division inside a logarithm into a simpler form. For example, when you see \( \log_a\left( \frac{x^2}{yz^3} \right) \), you can expand it by applying the quotient rule to become \( \log_a(x^2) - \log_a(yz^3) \).

This step is critical as it prepares the expression for further simplification using other logarithmic rules.

Product Rule

The product rule applies when a logarithm is dealing with a product of terms. Essentially, the logarithm of a product is the sum of the logarithms of the individual factors.

Using the product rule, for any base \( b \), this can be expressed as:

• \( \log_b(MN) = \log_b(M) + \log_b(N) \)

This means if you have an expression like \( \log_a(yz^3) \), it can be expanded to \( \log_a(y) + \log_a(z^3) \).

Applying the product rule is a helpful step because it breaks down the multiplication inside a logarithm into simpler additive terms. It clarifies the expression further and sets it up for using other log rules, like the power rule, to simplify terms raised to powers.

Power Rule

The power rule is extremely useful when you need to simplify expressions where a term inside a logarithm is raised to a power. The rule states that you can "move the power down" as a coefficient in front of the logarithm.

Mathematically, for a logarithm of any base \( b \), the power rule is expressed as:

• \( \log_b(M^n) = n \log_b(M) \)

This transformation is very effective. For example, in the expression \( \log_a(x^2) \), applying the power rule results in \( 2 \log_a(x) \). Similarly, \( \log_a(z^3) \) becomes \( 3 \log_a(z) \).

The power rule is the final touch in fully expanding complex logarithmic expressions, as it pulls out powers for easier manipulation and interpretation, giving you a cleaner, linear expression without unwieldy exponents.