Problem 31 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \sqrt[3]{\frac{x}{y}} $$

Step-by-Step Solution

Answer

\( \frac{1}{5} \log(x) - \frac{1}{5} \log(y) \)

Step 1: Converting root into its equivalent exponential form

To start with, we convert the root into its exponential equivalent, so \( \log \sqrt[5]{\frac{x}{y}} \) will convert into \( \log(\frac{x}{y})^{1/5} \)

Step 2: Applying the power rule of logarithms

We have the power rule of logarithms which states that for any positive number 'a', we have \( \log(a^n) = n \log a \). Now the next important step is to apply this rule to our equation. We will move the exponent in front of the log and we get \( \frac{1}{5} \log(\frac{x}{y}) \)

Step 3: Applying the quotient rule of logarithms

The quotient rule says that the logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator. So next, we apply the quotient rule to our function and end up with: \( \frac{1}{5} \log(x) - \frac{1}{5} \log(y) \)

Key Concepts

Properties of Logarithms

Logarithms have certain rules, known as properties, that help us manipulate and simplify logarithmic expressions easily. Understanding these properties can make complex logarithmic problems much simpler. - **Product Rule**: This states that the log of a product is the sum of the logs, i.e., \( \log(ab) = \log(a) + \log(b) \).- **Quotient Rule**: This states that the log of a quotient is the difference of the logs, i.e., \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \).- **Power Rule**: This states that the log of a number raised to a power is the exponent times the log of the base, i.e., \( \log(a^n) = n \log(a) \).These properties allow us to break down logarithmic expressions into more manageable parts, making calculations much easier. They are foundational to solving logarithmic equations and are widely used in mathematics.

Exponential Form

When dealing with logarithms, it’s often helpful to understand exponential form. Any expression involving a root can be rewritten as an expression involving exponents.For example, the nth root of a number \( a \), written as \( \sqrt[n]{a} \), can be expressed as \( a^{1/n} \).This is exactly what we have in step 1 of the problem, where \( \sqrt[5]{\frac{x}{y}} \) is rewritten as \((\frac{x}{y})^{1/5}\).Converting roots into exponential form is very useful. It allows us to use the power rule of logarithms which simplifies our calculations. Understanding and applying exponential form is crucial when working with complex logarithmic expressions.

Power Rule of Logarithms

The power rule of logarithms is a powerful tool that provides a convenient way to handle exponents within a logarithmic expression.This rule states that the logarithm of a number with an exponent can be expressed as the exponent multiplied by the logarithm of the base. Mathematically, it's expressed as \( \log(a^n) = n \log(a) \).In our exercise, after converting to exponential form, we have \( \log((\frac{x}{y})^{1/5}) \). Using the power rule, we can bring the exponent \( 1/5 \) in front, transforming it into \( \frac{1}{5} \log(\frac{x}{y}) \).This step greatly simplifies the expression because it isolates the exponent, making the next steps of simplification much easier. The power rule is integral to simplifying expressions with powers.

Quotient Rule of Logarithms

The quotient rule of logarithms is essential when dealing with divisions within a logarithm. It helps simplify the expression by breaking it down into the difference of two separate logs.The rule can be stated as: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \).In the given problem, after applying the power rule, our expression becomes \( \frac{1}{5} \log(\frac{x}{y}) \). Applying the quotient rule, we break this down into \( \frac{1}{5} \log(x) - \frac{1}{5} \log(y) \).This step effectively simplifies the logarithmic expression into a form that is easier to handle. The quotient rule is a key part of simplifying logarithmic expressions that involve fractions. It's an invaluable technique for breaking complex expressions into simpler components.