Problem 68 - A Graphical Approach to College Algebra

Question

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{4 p}{q}$$

Step-by-Step Solution

Answer

\(\log_{3}(4) + \log_{3}(p) - \log_{3}(q)\)

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms. Therefore, you can write: \[\log_{3} \left( \frac{4p}{q} \right) = \log_{3}(4p) - \log_{3}(q)\]

Step 2: Apply the Product Rule of Logarithms

The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms. Apply this rule to \(\log_{3}(4p)\):\[\log_{3}(4p) = \log_{3}(4) + \log_{3}(p)\]Now substitute this back into the equation from Step 1.

Step 3: Substitute and Simplify

Substitute the expression from Step 2 into the equation from Step 1:\[\log_{3} \left( \frac{4p}{q} \right) = \left( \log_{3}(4) + \log_{3}(p) \right) - \log_{3}(q)\]This simplifies to:\[\log_{3}(4) + \log_{3}(p) - \log_{3}(q)\]

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is an essential tool for simplifying complex logarithmic expressions. This rule tells us that if you have a logarithm of a fraction, you can express it as the difference between two separate logarithms. In everyday terms, this means you can "split" the log of a division into the subtraction of two logs.
For example, if you have the expression \( \log_{b} \left( \frac{x}{y} \right) \), according to the quotient rule, you can rewrite this as:

• \( \log_{b}(x) - \log_{b}(y) \)

In this context, the numerator (\(x\)) and the denominator (\(y\)) are separated, making the expression easier to manage and understand. By breaking down these expressions, complex problems become more straightforward and manageable. This rule is particularly helpful in solving logarithmic equations or in simplifying terms within an equation.

Product Rule of Logarithms

Understanding the product rule of logarithms is equally important when working with logarithmic expressions. The product rule states that the logarithm of a product is the sum of the logarithms for each factor involved.
Consider the logarithmic expression \( \log_{b}(xy) \). By using the product rule, you can break down this log into a simpler form:

• \( \log_{b}(x) + \log_{b}(y) \)

This indicates that when two or more numbers are multiplied together inside a logarithm, you can separate them by adding their individual logs. It's a handy tool to decompose and simplify logs without resorting to a calculator.
Using the product rule helps you to isolate each component of the product and potentially evaluate or simplify them further. It allows you to tackle logarithmic expressions more strategically, especially when dealing with calculations or transformations in algebra.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using rules like the quotient and product rules to rewrite the log in a more manageable form. The main goal is to make the expression easier to understand and solve.
When you have an expression like \( \log_{3} \left( \frac{4p}{q} \right) \), you can employ both rules for simplification. Start with the quotient rule to break it down:

• \( \log_{3}(4p) - \log_{3}(q) \)

Next, apply the product rule to further decompose \( \log_{3}(4p) \):

• \( \log_{3}(4) + \log_{3}(p) \)

Now, combining these steps gives you:

• \( \log_{3}(4) + \log_{3}(p) - \log_{3}(q) \)

This final form breaks the original expression into simpler components that are easier to interpret or use in further calculations. By using these rules, you remove the complexity from nested logarithmic expressions, paving the way for more effective problem-solving and mathematical reasoning.