Problem 86 - A Graphical Approach to College Algebra

Question

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\frac{1}{2} \log x-\frac{1}{3} \log y-2 \log z$$

Step-by-Step Solution

Answer

\( \log \left( \frac{\sqrt{x}}{\sqrt[3]{y} z^2} \right) \)

Step 1: Apply the Power Rule

The Power Rule states that \( a \log b = \log b^a \). We'll apply this to each term in the given expression:- \( \frac{1}{2} \log x \) becomes \( \log x^{1/2} = \log \sqrt{x} \).- \( \frac{1}{3} \log y \) becomes \( \log y^{1/3} = \log \sqrt[3]{y} \).- \( 2 \log z \) becomes \( \log z^2 \).

Step 2: Apply the Quotient Rule

The Quotient Rule states that \( \log a - \log b = \log \left(\frac{a}{b}\right) \). Now we have the expression:\( \log \sqrt{x} - \log \sqrt[3]{y} - \log z^2 \).Apply the rule to the last two terms:- \( \log \sqrt{x} - \log \sqrt[3]{y} - \log z^2 \) becomes \( \log \left( \frac{\sqrt{x}}{\sqrt[3]{y} \cdot z^2} \right) \).

Step 3: Combine into a Single Logarithm

Now the expression is in the form of a single logarithm:\[ \log \left( \frac{\sqrt{x}}{\sqrt[3]{y} \cdot z^2} \right) \]

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them incredibly useful for simplifying expressions involving exponents and division. These properties allow us to rewrite complex logarithmic expressions into simpler forms. There are three main properties that are commonly used:

• Product Rule: This rule states that the logarithm of a product is the sum of the logarithms of its factors: \( \log(a \times b) = \log a + \log b \).
• Quotient Rule: Similar to the product rule, the quotient rule states that the logarithm of a quotient is the difference of the logarithms: \( \log\left(\frac{a}{b}\right) = \log a - \log b \).
• Power Rule: This rule allows us to bring an exponent outside the log as a factor: \( \log(a^b) = b \log a \).

By using these properties, we can transform logarithmic expressions into simpler, equivalent forms. This is often useful for solving equations or simplifying expressions.

Power Rule

The Power Rule is a crucial tool for manipulating logarithms when they involve exponents or powers. To apply this rule, you'll need to understand that it effectively lets you step outside the logarithmic operation and deal with the power directly.Here's how it works: If you have an expression involving a logarithm with a coefficient, the power rule allows you to convert it into a single exponent form. For example:- \( a \log b \) becomes \( \log b^a \).In practical terms, this means:

• If you see \( \frac{1}{2} \log x \), using the power rule, it becomes \( \log x^{1/2} = \log \sqrt{x} \). This conversion makes it easier to work with the expression because the power rule "removes" the coefficient away from the logarithm.
• Similarly, \( 2 \log z \) turns into \( \log z^2 \).

By applying this rule, you can start combining and reducing logarithmic terms, leading to a simpler overall expression.

Quotient Rule

The Quotient Rule is another fundamental property of logarithms that helps in simplifying expressions involving divisions within logarithms. This rule is particularly helpful when you are working with expressions that involve subtraction between different logarithmic terms.The rule states:

• \( \log a - \log b = \log\left(\frac{a}{b}\right) \).

This transformation simplifies expressions since it allows you to express a difference of logs as a single log involving a fraction.For example, suppose you have the logarithmic expression \( \log \sqrt{x} - \log \sqrt[3]{y} - \log z^2 \). By applying the quotient rule, it can be rewritten as:

• \( \log\left(\frac{\sqrt{x}}{\sqrt[3]{y} \cdot z^2}\right) \).

This final single logarithmic expression includes all the components from the original expression, but it is simplified to one manageable term. The Quotient Rule is invaluable for such transformations, as it allows the subtraction of logarithms to succinctly capture the relationship between different components as a division.