Problem 17 - Precalculus : Building Concepts and Connections

Question

In Exercises \(15-20,\) use the properties of logarithms to simplify each expression by eliminating all exponents and radicals. Assume that \(x, y > 0\). $$\log \sqrt[3]{x} \sqrt[4]{y}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \((1/3)\log (x) + (1/4)\log (y)\)

Step 1: Express roots as fractional exponents

Rewrite the cube root and fourth root of x and y respectively as a fractional exponent. The cube root of \(x\) is \(x^{1/3}\) and the fourth root of \(y\) is \(y^{1/4}\). So, the expression becomes: \(\log (x^{1/3} . y^{1/4})\)

Step 2: Use log property to split the log expression

Using the logarithmic property \(\log(ab) = \log(a) + \log(b)\), the expression inside the log can be split: \(\log (x^{1/3}) + \log (y^{1/4})\)

Step 3: Use log property to eliminate exponents

Using another logarithmic property, \(\log(a^p) = p\log(a)\), move the exponents in front of the log to eliminate them: \((1/3)\log (x) + (1/4)\log (y)\)

Key Concepts

Properties of Logarithms

The properties of logarithms are useful tools in algebra, especially for simplifying complex expressions. One powerful property is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as \(\log(ab) = \log(a) + \log(b)\). This is incredibly handy when you have logarithms with multiplied expressions inside.
There's also the power rule. This tells us how to handle logarithms of expressions that include exponents. Specifically, \(\log(a^p) = p\cdot\log(a)\). The exponent can be brought down in front as a multiplier, which often simplifies the expression significantly.
These properties let us break down and manipulate algebraic expressions in manageable ways, essential for solving equations efficiently. With practice, applying these rules becomes a straightforward task making logs much easier to deal with.

Fractional Exponents

Fractional exponents are another way to represent roots in mathematics. Understanding this concept is crucial as it provides a consistent method for handling roots and powers. For example, the square root of a number is the same as raising that number to the power of \(\frac{1}{2}\). Similarly, the cube root of \(x\) is represented as \(x^{1/3}\) and the fourth root of \(y\) is represented as \(y^{1/4}\).
These notations make it easier when dealing with expressions like \(\sqrt[3]{x}\sqrt[4]{y}\) because you can write it in a uniform format as \(x^{1/3}\cdot y^{1/4}\).
Using fractional exponents aligns operations with the rules of exponents, such as multiplying, which simplifies complex expressions. This technique ensures consistency across various mathematical tasks and prepares us to handle even more advanced topics.

Simplifying Expressions

Simplifying expressions is about making them as neat and manageable as possible without changing their value. This often involves removing exponents, condensing terms, or using properties of operations, such as distributive or associative properties.
In exercises involving roots or logarithms, it's common to replace roots with fractional exponents, apply logarithmic properties, and condense terms as much as possible. For instance, in logarithmic expressions like \(\log(x^{1/3} \cdot y^{1/4})\), simplifying might involve splitting it into \(\log(x^{1/3}) + \log(y^{1/4})\) using the product rule.
Additionally, by transforming \(\log(x^{1/3})\) into \(\frac{1}{3}\log(x)\) using the power rule, we further simplify the expression.
The goal is always to transform the expression into its simplest form, making it easier to interpret or solve within the context of a problem, improving efficiency, and reducing complexity.