Problem 41 - Precalculus

Question

Write each expression as a single logarithm. $$\frac{1}{2} \log _{b} x+\frac{2}{3} \log _{b} y$$

Step-by-Step Solution

Answer

\( \log_b (x^{1/2}y^{2/3}) \)

Step 1: Apply Power Rule

Begin by applying the power rule of logarithms, which states that \( a \cdot \log_b M = \log_b M^a \). Using this rule, transform each term of the expression. The first term becomes \( \log_b (x^{1/2}) \) and the second term becomes \( \log_b (y^{2/3}) \).

Step 2: Use the Product Rule

Now that both logarithmic terms have been converted, apply the product rule of logarithms, which states \( \log_b M + \log_b N = \log_b (M \cdot N) \). Substitute the expressions from Step 1 into this rule to combine the terms into a single logarithm: \( \log_b (x^{1/2}y^{2/3}) \).

Key Concepts

Power Rule

The power rule in logarithms is a useful tool, often simplifying expressions where a constant multiplies a logarithm. It enables us to rewrite expressions in a more manageable form.

The rule states that if you have a logarithm of a number raised to a power, such as \( a \cdot \log_b M \), it can be rewritten as \( \log_b M^a \). This is a consequence of the properties of exponents within a logarithm that allows the movement of the exponent to the outside of the logarithmic function:

• For example, \( \frac{1}{2} \log_b x \) becomes \( \log_b (x^{1/2}) \)
• Similarly, \( \frac{2}{3} \log_b y \) becomes \( \log_b (y^{2/3}) \)

By using the power rule, you can simplify each part of an equation involving logarithms, setting up for further simplifications. This approach is particularly useful as a preliminary step before applying other rules, like the product rule.

Product Rule

The product rule of logarithms provides a method for combining two logarithms into a single expression. This is achieved by translating the sum of two logs into the logarithm of a product, which greatly simplifies the expression.

According to the product rule, \( \log_b M + \log_b N = \log_b (M \cdot N) \).

• Using the expressions provided from the power rule, \( \log_b (x^{1/2}) + \log_b (y^{2/3}) \) can be consolidated using the product rule
• The resulting expression is \( \log_b (x^{1/2} \cdot y^{2/3}) \), simplified into a single logarithm

This rule saves time and reduces errors by transforming a sum of logs into a straightforward multiplication inside a single logarithm, crucial for progressing in more complex computations.

Expression Simplification

Simplifying logarithmic expressions involves transforming them into an easier-to-read and compute form. This is particularly helpful in algebra and calculus, where complex expressions frequently need to be handled efficiently.

Expression simplification often combines multiple rules and properties of logarithms:

• Start by addressing individual parts of the expression with the power rule
• Then, blend the terms using the product or quotient rules, as applicable

The original problem \( \frac{1}{2} \log_b x + \frac{2}{3} \log_b y \) was simplified by first using the power rule on each term, then the product rule to consolidate them.

With practice, these techniques become second nature, equipping students to tackle more intricate logarithmic challenges with confidence.