Problem 53 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 2 \log _{b} x+3 \log _{b} y $$

Step-by-Step Solution

Answer

The condensed logarithmic expression is \( \log_b (x^2y^3)\)

Step 1: Identify the properties of logarithms to be used

The properties of logarithms that will be used here are: \n1) \(a \log_b m = \log_b m^a\)\n2) \(\log_b m + \log_b n = \log_b (mn)\)

Step 2: Apply the power rule to each term

Using rule 1, the expression can be rewritten as:\n \( \log_b x^2 + \log_b y^3\)

Step 3: Apply the product rule

Using rule 2, the expression can be rewritten as a single logarithm:\n \( \log_b (x^2y^3)\)

Key Concepts

Properties of Logarithms

Logarithms possess unique properties that make it easier to manipulate and simplify expressions. Let's explore a couple of crucial properties used in the exercise.

- **Power Rule**: The power rule states that multiplying a logarithm by a coefficient can be expressed as raising the logarithm's argument to that coefficient. Mathematically, it's written as: \( a \log_b m = \log_b m^a \).

This means you can "move" a coefficient into the logarithmic argument as an exponent. - **Product Rule**: This rule helps to combine two logarithms with the same base that are added together. It states: \( \log_b m + \log_b n = \log_b (mn) \).

Here, you multiply the terms inside the logarithms' arguments. These properties simplify complex logarithmic expressions into a single, more manageable expression. Understanding these rules is vital when working with logarithms in mathematics.

Logarithmic Expressions

A logarithmic expression contains one or several logarithms. These expressions can be simplified using the properties of logarithms. For example, in the problem provided, you have the expression \( 2 \log_b x + 3 \log_b y \).

By utilizing the properties, you can rewrite and simplify it. The expression contains multiple logarithmic terms and coefficients. By knowing how to apply the power rule, the coefficients become parts of the exponent for each logarithmic term, such as turning \( 2 \log_b x \) into \( \log_b x^2 \).

Similarly, \( 3 \log_b y \) becomes \( \log_b y^3 \). Using the product rule further simplifies these into a single logarithmic expression: \( \log_b (x^2y^3) \).

Learning to simplify such expressions is invaluable, not just for this exercise but also for solving logarithmic equations.

Condensing Logarithms

Condensing logarithms means using logarithmic properties to combine several logarithmic terms into one. This process is beneficial because it simplifies calculations and makes expressions easier to handle.

Consider the expression from the exercise: \( 2 \log_b x + 3 \log_b y \). To condense:
- Apply the power rule first, transforming it into \( \log_b x^2 + \log_b y^3 \).- Then, use the product rule to combine it into a single logarithm: \( \log_b (x^2y^3) \).

After condensing, the expression is now simplified into one term, making it more elegant and easier to evaluate if necessary. Condensing logarithms is an essential skill for simplifying complex logarithmic expressions in higher-level math topics.