Problem 74 - Algebra for College Students

Question

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} x^{2} y^{3} $$

Step-by-Step Solution

Answer

\( \log_{b}(x^2 y^3) = 2 \log_{b}(x) + 3 \log_{b}(y) \).

Step 1: Apply the Product Rule for Logarithms

The product rule for logarithms states that the logarithm of a product is the sum of the logarithms of the factors. Apply this rule to the expression: \[ \log_{b}(x^2 y^3) = \log_{b}(x^2) + \log_{b}(y^3). \]

Step 2: Apply the Power Rule for Logarithms

The power rule for logarithms states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the base number. Apply this rule to each term separately: \[ \log_{b}(x^2) = 2 \log_{b}(x) \] and \[ \log_{b}(y^3) = 3 \log_{b}(y). \]

Step 3: Combine the Simplified Expressions

Combine the simplified expressions from Step 2 to express the original logarithmic expression as a sum of simpler terms:\[ \log_{b}(x^2 y^3) = 2 \log_{b}(x) + 3 \log_{b}(y). \]

Key Concepts

Product Rule for Logarithms

The product rule for logarithms is a handy tool when dealing with the logarithm of a product of numbers. Imagine you have two numbers multiplied together, like in the expression \( \log_{b}(x^2 y^3) \). The product rule tells us that the logarithm of a product is the same as adding the logarithm of each factor. This makes our job simpler.

For example, using the product rule, \( \log_{b}(x^2 y^3) \) can be broken down into:

• \( \log_{b}(x^2) \) — the logarithm of \( x^2 \)
• \( \log_{b}(y^3) \) — the logarithm of \( y^3 \)

When applied, the product rule would transform our original expression into: \[ \log_{b}(x^2) + \log_{b}(y^3). \]This method is essential for simplifying more complex expressions involving multiple factors.

Power Rule for Logarithms

The power rule for logarithms simplifies expressions where logarithms have exponents. Simply put, this rule allows you to "bring down" the exponent in front of the logarithm.

This rule states: If you have \( \log_{b}(x^n) \), it can be rewritten as \( n \log_{b}(x) \).

For instance, applying this to our expression \( \log_{b}(x^2) \) turns into:

• \( 2 \log_{b}(x) \)

And similarly, \( \log_{b}(y^3) \) becomes:

• \( 3 \log_{b}(y) \)

By using the power rule, we're essentially unraveling the power into a multiplication outside the logarithm. It simplifies solving and understanding logarithmic expressions, especially when multiple factors involve exponents.

Logarithmic Identities

Logarithmic identities are a set of rules that help us manipulate and simplify logarithmic expressions. They are crucial when you want to break down expressions into easier or more familiar terms.

One of the key identities we used here is combining the results from applying the product and power rules.

In our original problem, we concluded that \( \log_{b}(x^2 y^3) \) can be expressed more simply as:

• \( 2 \log_{b}(x) + 3 \log_{b}(y) \)

Underpinning this breakdown is the identity that if you understand how to apply the product and power rules effectively, you can decompose complex logarithmic statements into a series of simpler parts. These identities are fundamental in algebra and calculus for making challenging problems more tractable.