Problem 32 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \ln \sqrt[3]{3 r^{2} s} $$

Step-by-Step Solution

Answer

\( \frac{1}{3} \ln(3) + \frac{2}{3} \ln(r) + \frac{1}{3} \ln(s) \)

Step 1: Apply the Power Rule

The expression is given as \( \ln \sqrt[3]{3 r^{2} s} \). Recognize that the cube root can be expressed as a fractional power: \( \sqrt[3]{3 r^{2} s} = (3 r^{2} s)^{1/3} \). Use the power rule for logarithms, which states \( \ln(a^b) = b \cdot \ln(a) \), to rewrite the expression: \( \ln \left( (3r^2 s)^{1/3} \right) = \frac{1}{3} \ln(3r^2 s) \).

Step 2: Apply the Product Rule

The expression inside the logarithm is \( \ln(3r^2 s) \). By using the product rule for logarithms, which states \( \ln(abc) = \ln(a) + \ln(b) + \ln(c) \), we can expand this to: \( \ln(3) + \ln(r^2) + \ln(s) \).

Step 3: Apply the Power Rule Again

In \( \ln(r^2) \), apply the power rule for logarithms, \( \ln(a^b) = b \cdot \ln(a) \), which gives: \( \ln(r^2) = 2 \cdot \ln(r) \). Substitute this into the expression to get \( \ln(3) + 2 \ln(r) + \ln(s) \).

Step 4: Combine and Simplify the Expression

Now substitute back into the expression obtained from Step 1: \( \frac{1}{3} (\ln(3) + 2 \ln(r) + \ln(s)) \). Distribute \( \frac{1}{3} \) through the expanded expression to get \( \frac{1}{3} \ln(3) + \frac{2}{3} \ln(r) + \frac{1}{3} \ln(s) \).

Key Concepts

Power Rule

The Power Rule in logarithms is a powerful tool that helps simplify expressions raised to an exponent within a logarithm. Its basic form is given by: \[ \ln(a^b) = b \cdot \ln(a) \] This rule tells us that we can "pull down" the exponent in front of the logarithm as a multiplier, making complex expressions more manageable.

• First, recognize powers within logarithmic expressions.
• Apply the rule to move the exponent outside.

In the given exercise, the expression \( \ln \sqrt[3]{3 r^{2} s} \) involves a cube root, which we converted to a power: \( (3 r^2 s)^{1/3} \). This transformation allows us to apply the Power Rule, resulting in \( \frac{1}{3} \ln(3 r^2 s) \). This simplifies handling roots and fractional exponents effectively in logarithmic equations.

Product Rule

The Product Rule aids in breaking down complex logarithmic expressions involving products of multiple terms inside a logarithm. The rule states: \[ \ln(abc) = \ln(a) + \ln(b) + \ln(c) \] This tells us to split the expression into a sum of the logarithms of individual factors.

• Identify products within the logarithm.
• Split them into separate logarithmic terms.

In our example, with \( \ln(3 r^2 s) \), the expression can be broken down to \( \ln(3) + \ln(r^2) + \ln(s) \). By doing so, each part becomes easier to work with individually. The Product Rule is particularly useful for handling multiplicative components within logarithms, making further simplifications straightforward.

Logarithmic Expansion

Logarithmic Expansion refers to the process of transforming a single, often complicated, logarithmic expression into a sum of simpler terms using logarithmic rules. This step-by-step expansion employs both the Power Rule and the Product Rule.

• Start by recognizing complex forms like products and powers.
• Systematically apply appropriate logarithmic laws to expand the expression.

In our task, we first applied the Power Rule to bring any exponents outside the logarithm: \( \frac{1}{3} \ln(3 r^2 s) \). Next, using the Product Rule, we expanded the logarithm: \( \ln(3) + \ln(r^2) + \ln(s) \). Finally, by applying the Power Rule again on \( \ln(r^2) \), we achieved a fully expanded form: \( \frac{1}{3} \ln(3) + \frac{2}{3} \ln(r) + \frac{1}{3} \ln(s) \).
Through logarithmic expansion, we transform initial complex expressions into manageable parts that are easy to handle and solve.