Problem 77 - 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. $$\log _{a} x+\log _{a} y-\log _{a} m$$

Step-by-Step Solution

Answer

\( \log_a \left( \frac{xy}{m} \right) \)

Step 1: Apply the Product Rule for Logarithms

The product rule of logarithms states that \( \log_b (mn) = \log_b m + \log_b n \). Apply this to combine the first two terms: \( \log_a x + \log_a y = \log_a (xy) \).

Step 2: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \). Use this to combine the result from Step 1 with the third term: \( \log_a (xy) - \log_a m = \log_a \left( \frac{xy}{m} \right) \).

Step 3: Express as a Single Logarithm

The final expression is \( \log_a \left( \frac{xy}{m} \right) \). This is a single logarithm with a coefficient of 1.

Key Concepts

Product Rule

The product rule for logarithms is a very handy tool when you need to add logarithms together. If you have the expression \( \log_b m + \log_b n \), you can combine these into a single term: \( \log_b (mn) \). This means multiplying the arguments of the logs inside one log because logarithms are essentially exponents, and this rule reflects the property of exponents where adding them means multiplying the base numbers.

This rule comes into play when you're dealing with problems that require you to simplify or solve logarithmic expressions, much like in our example where \( \log_a x + \log_a y \) becomes \( \log_a (xy) \).

• Remember to use the same base for the logs you're adding.
• This rule primarily helps in reducing the complexity of the expression by combining terms.

As you encounter more logarithmic expressions, the product rule will often be your first tool to simplify the expression and make your calculations easier.

Quotient Rule

The quotient rule for logarithms is another useful tool, especially when you deal with subtraction of logarithm expressions. This rule states that if you subtract logarithms with the same base, like \( \log_b m - \log_b n \), you can rewrite this as \( \log_b \left( \frac{m}{n} \right) \).

Just as dividing exponents translates into subtracting them, the quotient rule reflects this in logarithmic terms. When you have an expression like \( \log_a (xy) - \log_a m \), you're subtracting a log and can simplify it as \( \log_a \left( \frac{xy}{m} \right) \).

• Ensure the logarithms have the same base before applying this rule.
• This method helps condense multiple logs into a single log expression.

This simplification is crucial in advanced mathematical problems involving logarithms, aiding in both solving and understanding the relations between quantities.

Properties of Logarithms

Understanding the properties of logarithms is key to simplifying complex expressions involving logs. The product and quotient rules are just two of these properties. They greatly assist in combining and reducing expressions, making calculations manageable and straightforward. Here are a few broad properties:

• The power rule, which states \( \log_b (m^n) = n\log_b m \), helps when a term is raised to a power.
• Change of base property, allowing the change of bases for ease: \( \log_b m = \frac{\log_k m}{\log_k b} \).
• The log of a base to itself is always one, \( \log_b b = 1 \), providing a useful simplification.

With the exercise example, these properties reduce the task into manageable steps by applying the product and quotient rules. Mastery of logarithm properties will simplify complex algebraic problems, paving the way for understanding more advanced topics in mathematics.