Problem 76 - Intermediate Algebra

Question

For Problems \(69-80\), express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} x^{\frac{2}{3}} y^{\frac{3}{4}} $$

Step-by-Step Solution

Answer

\( \frac{2}{3} \log_b x + \frac{3}{4} \log_b y \)

Step 1: Identify the Logarithm Expression

We are given the expression \( \log_b\left(x^{\frac{2}{3}} y^{\frac{3}{4}}\right) \). Our goal is to break this down into a sum or difference of simpler logarithms using logarithmic properties.

Step 2: Use the Product Property of Logarithms

Recall the product property of logarithms: \( \log_b(uv) = \log_b u + \log_b v \). Apply this to the expression \( \log_b\left(x^{\frac{2}{3}} y^{\frac{3}{4}}\right) \), which gives: \[ \log_b\left(x^{\frac{2}{3}}\right) + \log_b\left(y^{\frac{3}{4}}\right) \].

Step 3: Apply the Power Property of Logarithms

Use the power property of logarithms: \( \log_b(u^n) = n\log_b u \). Apply this to each term: \[ \frac{2}{3} \log_b x + \frac{3}{4} \log_b y \].

Step 4: Combine the Results

The expression \( \log_b\left(x^{\frac{2}{3}} y^{\frac{3}{4}}\right) \) is expressed as the sum of two simpler logarithmic terms: \[ \frac{2}{3} \log_b x + \frac{3}{4} \log_b y \].

Key Concepts

Product Property of Logarithms

The product property of logarithms is highly useful for simplifying logarithmic expressions. It states that the logarithm of a product is the sum of the logarithms of the factors. More formally, for any positive real numbers \( u \) and \( v \), and a base \( b \), this property is expressed as \( \log_b(uv) = \log_b u + \log_b v \). This is particularly helpful when you want to break down complex logarithmic expressions involving multiplication into simpler parts.

Let's apply this to the given problem. The expression \( \log_b(x^{\frac{2}{3}} y^{\frac{3}{4}}) \) involves a product inside the logarithm. According to the product property, we can split this into \( \log_b(x^{\frac{2}{3}}) + \log_b(y^{\frac{3}{4}}) \). This initial step makes the problem much easier to handle, setting the stage for further simplification.

Power Property of Logarithms

The power property of logarithms allows us to deal with exponents within logarithms seamlessly. It tells us that for any positive real number \( u \), base \( b \), and exponent \( n \), the expression \( \log_b(u^n) = n \log_b u \) holds true. This property is particularly helpful when simplifying expressions where variables are raised to powers.

Returning to our expression post-product step: \( \log_b(x^{\frac{2}{3}}) + \log_b(y^{\frac{3}{4}}) \), we can apply the power property to each logarithm. For \( \log_b(x^{\frac{2}{3}}) \), using the power property gives us \( \frac{2}{3} \log_b x \). Similarly, \( \log_b(y^{\frac{3}{4}}) \) transforms into \( \frac{3}{4} \log_b y \). These transformations further break down the complex expression into a simpler, more manageable form.

Simplifying Logarithmic Expressions

Finally, simplifying logarithmic expressions often involves using both the product and power properties in tandem. We start with a complex expression and use these properties step-by-step to break it down into its simplest form.

• Start with the product property: Split any multiplied components within the logarithm into separate terms.
• Next, apply the power property: Simplify terms with exponents by bringing the exponent in front as a coefficient.

In our solved problem, we began with \( \log_b(x^{\frac{2}{3}} y^{\frac{3}{4}}) \). Using the properties as detailed, we ended up with \( \frac{2}{3} \log_b x + \frac{3}{4} \log_b y \). This clear expression is now composed of simplified terms, each understandable by itself.

Mastering these strategies equips you with powerful tools to tackle a wide range of logarithmic expressions, making complex mathematical scenarios more approachable and less intimidating. Simplification through these properties not only achieves a more workable form but also aids in deeper understanding of logarithmic behaviors.