Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. \(\log (x)-\frac{1}{2} \log (y)+3 \log (z)\)

Step-by-Step Solution

Answer

\(\log\left(\frac{xz^3}{\sqrt{y}}\right)\)

Step 1: Identify Logarithm Properties

Before condensing the expression, recall the key properties of logarithms: the power rule, the product rule, and the quotient rule. The power rule states that \(a \log b = \log(b^a)\). The product rule states that \(\log a + \log b = \log(ab)\). The quotient rule states that \(\log a - \log b = \log\left(\frac{a}{b}\right)\).

Step 2: Apply Power Rule

Use the power rule to simplify each term where applicable. Rewrite each term so that any coefficients in front of the logarithms become exponents.- The term \(-\frac{1}{2} \log(y)\) becomes \(\log(y^{-1/2})\).- The term \(3 \log(z)\) becomes \(\log(z^3)\).So the expression becomes: \(\log(x) + \log(z^3) - \log(y^{1/2})\).

Step 3: Apply Product Rule

Combine the logarithms using the product rule for the addition part of the expression. Add the terms \(\log(x)\) and \(\log(z^3)\).The expression becomes: \(\log(x \cdot z^3) - \log(y^{1/2})\).

Step 4: Apply Quotient Rule

Use the quotient rule to combine the remaining terms into a single logarithm. The expression is currently in the form \(\log(a) - \log(b)\) where \(a = x \cdot z^3\) and \(b = y^{1/2}\).Thus, combine them into: \(\log\left(\frac{x \cdot z^3}{y^{1/2}}\right)\).

Step 5: Final Expression

The condensed expression is \(\log\left(\frac{xz^3}{\sqrt{y}}\right)\), where we used that \(y^{1/2} = \sqrt{y}\).

Key Concepts

Power Rule

The power rule in logarithms is a fundamental property that allows you to bring coefficients of logarithms as exponents within the log expression itself. Imagine you have the expression \(a \log(b)\); the power rule transforms it into \(\log(b^a)\). This property can simplify complex expressions, making calculations easier and more manageable.

Here's how it works:

• Take \(-\frac{1}{2} \log(y)\), applying the power rule changes it to \(\log(y^{-1/2})\).
• Similarly, for \(3 \log(z)\), it becomes \(\log(z^3)\) using the power rule.

By turning coefficients into exponents, you can manage logarithmic expressions better and prepare for other rules such as the product and quotient rules.

Product Rule

The product rule for logarithms is your tool for combining logarithmic expressions through addition. When you have multiple logs being added, like \(\log(a) + \log(b)\), the product rule lets you merge them into \(\log(ab)\). This makes the expression more compact and straightforward.

Consider these simplified terms from the power rule:

• \(\log(x)\)
• \(\log(z^3)\)

Using the product rule, these combine into \(\log(x \cdot z^3)\).

This process of merging helps reduce complexity, allowing for a single logarithm expression before applying the next rules.

Quotient Rule

The quotient rule of logarithms is used to manage expressions with subtraction between logarithms. It allows you to condense \(\log(a) - \log(b)\) into a single expression: \(\log\left(\frac{a}{b}\right)\). This rule is particularly helpful in reducing expressions by removing extra logarithms.

To see this in action, apply it to the already simplified form from previous steps:

• \(\log(x \cdot z^3)\)
• \(\log(y^{1/2})\)

Using the quotient rule, these expressions join succinctly to become \(\log\left(\frac{x \cdot z^3}{y^{1/2}}\right)\),resulting in a neater, single logarithmic expression.

This technique is often the final step in condensing logarithmic expressions, bringing everything together efficiently.