Problem 79 - 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. $$2 \log _{m} a-3 \log _{m} b^{2}$$

Step-by-Step Solution

Answer

\( \log_m \left(\frac{a^2}{b^6}\right) \)

Step 1: Apply the Power Rule

The power rule of logarithms states that for any logarithm \( \log_b(x^n) = n \cdot \log_b(x) \). We'll apply this rule to both terms in the expression: \( 2 \log_m a \) and \( 3 \log_m b^2 \).- For \( 2 \log_m a \): Rewrite as \( \log_m (a^2) \). - For \( 3 \log_m b^2 \): Rewrite as \( \log_m (b^{6}) \).

Step 2: Apply the Quotient Rule

The quotient rule of logarithms states that \( \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) \). Apply this rule to combine the two log expressions:Combine \( \log_m (a^2) - \log_m (b^{6}) \) into a single logarithm using the quotient rule: \( \log_m \left(\frac{a^2}{b^6}\right) \).

Step 3: Simplify the Expression

After applying the rules, the expression becomes \( \log_m (\frac{a^2}{b^6}) \). This is the simplified form of the original expression written as a single logarithm with coefficient 1.

Key Concepts

Properties of Logarithms

The properties of logarithms are a set of rules that allow us to simplify logarithmic expressions. Understanding these properties helps solve complex problems. The main properties are:

• The Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• The Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• The Power Rule: \( \log_b(x^n) = n \cdot \log_b(x) \)

Each rule becomes a tool to rewrite and combine logarithmic expressions into simpler forms.
The goal is often to collapse multiple logarithms into a single logarithm.

Power Rule

The power rule is one of the most useful properties for simplifying logarithms. It states that the logarithm of a power can be rewritten as a product: \( \log_b(x^n) = n \cdot \log_b(x) \).
Let's break this down. If you have a logarithm of a term raised to a power, you can move the exponent down in front of the logarithm as a factor.
This is extremely useful in making expressions simpler, especially when needing to combine or manipulate terms.

Consider the expression given in our exercise:

• For \( 2 \log_m a \), the power rule rewrites this as \( \log_m (a^2) \).
• For \( 3 \log_m b^2 \), the power rule allows us to write this as \( \log_m (b^6) \).

This power rule application is the first step in our simplification process.

Quotient Rule

The quotient rule is another powerful property of logarithms. It states that the difference of two logarithms with the same base can be rewritten as a single logarithm of a quotient: \( \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right) \).
This rule is crucial when you have subtraction between logs and want to simplify them into one expression.

In the context of the exercise, after applying the power rule, we end up with the expression \( \log_m (a^2) - \log_m (b^6) \). By using the quotient rule, we can merge these into a single logarithm:

• Combine to form \( \log_m \left(\frac{a^2}{b^6}\right) \).

The quotient rule thus simplifies the expression significantly, which is the key to solving the exercise.

Simplifying Expressions

Simplifying expressions in logarithms usually involves rewriting them into a more compact form. This typically means using properties of logarithms to combine multiple terms into a single logarithmic term.
The process generally follows these steps:

• Apply the power rule to bring exponents in front of the log.
• Use the quotient or product rule to combine terms into a single logarithm.

In our exercise, we first applied the power rule to write each term as a single log expression. Then, the quotient rule was used to combine these into \( \log_m \left(\frac{a^2}{b^6}\right) \).
This final form is much simpler and meets the requirement of having a single log with coefficient 1, achieving the objective of the exercise.