Problem 85 - 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. $$3 \log x-4 \log y$$

Step-by-Step Solution

Answer

\(\log\left(\frac{x^3}{y^4}\right)\)

Step 1: Identify Logarithmic Properties

The given expression is \(3 \log x - 4 \log y\). We will use two properties of logarithms: the Power Rule and the Quotient Rule. The Power Rule states \( a \log b = \log(b^a) \) and the Quotient Rule states \( \log a - \log b = \log\left(\frac{a}{b}\right) \).

Step 2: Apply the Power Rule

Apply the Power Rule to both terms: \(3 \log x\) becomes \(\log(x^3)\) and \(4 \log y\) becomes \(\log(y^4)\). Now, rewrite the expression as \(\log(x^3) - \log(y^4)\).

Step 3: Apply the Quotient Rule

Use the Quotient Rule to rewrite \(\log(x^3) - \log(y^4)\) as a single logarithm: \(\log\left(\frac{x^3}{y^4}\right)\). This expression is now in the desired form, where it is a single logarithm with the coefficient of 1.

Key Concepts

Power Rule

The Power Rule is one of the fundamental properties of logarithms that simplifies the multiplication of a logarithm by a constant. When you see an expression like \( a \log b \), think of it as repeated addition of \( \log b \), which can be transformed into a more compact form. By the Power Rule, this expression becomes \( \log(b^a) \).

This transformation is particularly useful when you are dealing with large exponents or need to simplify expressions for further manipulation.

• The Power Rule helps in reducing complexity by transforming the coefficient into an exponent.
• It makes expressions easier to manage, especially in algebraic manipulations involving multiple logarithms.

For example, in our original exercise, the task involves transforming \(3 \log x\) into \(\log(x^3)\). This application of the Power Rule not only simplifies but also prepares the expression for further operations, like the Quotient Rule.

Quotient Rule

The Quotient Rule is another vital property of logarithms, allowing you to simplify differences between two logarithms. Specifically, when you see an expression like \( \log a - \log b \), the Quotient Rule lets you express it as a single logarithm: \( \log\left(\frac{a}{b}\right) \).
This rule is incredibly useful when you want to rewrite complex logarithmic expressions as a single log term, simplifying calculations and interpretations.

• The Quotient Rule transforms subtraction into division within the logarithmic function.
• It aids in seamlessly merging two separate logarithms into one entity, thus making the expression straightforward and concise.

For instance, in the step-by-step solution, after applying the Power Rule to convert \(3 \log x - 4 \log y\) into \(\log(x^3) - \log(y^4)\), we then apply the Quotient Rule to compile it into a single logarithmic statement: \(\log\left(\frac{x^3}{y^4}\right)\). This transforms the expression without any coefficient, fulfilling the exercise's requirement.

Logarithmic Expressions

Logarithmic expressions might seem daunting initially, but they are just another way to represent exponents and multiplication. At the core, these expressions make many mathematical operations, especially those involving large numbers or complex formulas, easier to understand and work with.

A logarithmic expression generally consists of a logarithm function acting on a base. Simplifying these expressions involves using logarithmic properties like the Power and Quotient Rules, which can help you manage and transform them.

• Logarithmic expressions can represent repetitive multiplicative processes as sophisticated yet simplified forms.
• Understanding how to manipulate these expressions using various logarithmic properties enables more effective problem-solving and computational efficiency.

In the original exercise, the ultimate goal was simplifying \(3 \log x - 4 \log y\) into one cohesive logarithmic form \(\log\left(\frac{x^3}{y^4}\right)\). This demonstrates not only the utility of logarithmic properties but also highlights the elegance of transforming potentially complicated algebraic expressions into simpler forms.