Problem 82 - Algebra for College Students

Question

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \log _{b} x+\log _{b} y-\log _{b} z $$

Step-by-Step Solution

Answer

\(\log_b\left(\frac{xy}{z}\right)\)

Step 1: Understand the Logarithm Properties

In order to combine the given logarithms into a single expression, we need to understand the fundamental properties of logarithms:1. The Product Rule: \(\log_b M + \log_b N = \log_b(MN)\)2. The Quotient Rule: \(\log_b M - \log_b N = \log_b\left(\frac{M}{N}\right)\)

Step 2: Apply the Product Rule

Using the Product Rule on the expression \(\log_b x + \log_b y\), we combine these terms under a single logarithm:\(\log_b x + \log_b y = \log_b(xy)\).

Step 3: Apply the Quotient Rule

Now use the Quotient Rule on \(\log_b(xy) - \log_b z\), combining these into a single logarithm:\(\log_b(xy) - \log_b z = \log_b\left(\frac{xy}{z}\right)\).

Step 4: Write the Final Expression

The expression \(\log_b x + \log_b y - \log_b z\) simplifies to a single logarithmic expression:\(\log_b\left(\frac{xy}{z}\right)\).

Key Concepts

Product Rule of Logarithms

Understanding the product rule of logarithms is crucial when simplifying expressions. The product rule states that when you add two logarithms with the same base, you can combine them by multiplying the arguments. Specifically, the formula is:

• \(\log_b M + \log_b N = \log_b (MN)\)

Think of this rule as expanding multiplication into addition, which is the essence of logarithms as the inverse of exponents. For instance, if you have \(\log_b x + \log_b y\), you can rewrite this as \(\log_b(xy)\).
By applying this rule, you can simplify complex logarithmic expressions to make calculations easier. It's as if you're compressing or bundling multiplication under one logarithmic operation.

Quotient Rule of Logarithms

The quotient rule of logarithms helps simplify expressions involving division inside logarithms. When you subtract two logarithms with the same base, you can combine them by using division within a single log. The formula is as follows:

• \(\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)\)

This rule is quite helpful when dealing with divisions and is analogous to how the product rule combines multiplications through addition, but in reverse. For example, given \(\log_b(xy) - \log_b z\), applying the quotient rule helps you to simplify it to \(\log_b \left(\frac{xy}{z}\right)\).
Utilizing this rule means you can manage expressions in a more compact form, making it easier to work through the operations without overwhelming calculations.

Properties of Logarithms

The properties of logarithms include a set of rules that aid in simplifying logarithmic expressions. These include the product rule, quotient rule, and other key properties such as the power rule that are essential in the manipulation of logarithmic terms:

• Product Rule: Adds logarithms to combine them under multiplication.
• Quotient Rule: Subtracts logarithms to combine them under division.
• Power Rule: \(\log_b(M^n) = n \cdot \log_b(M)\) - a property that brings exponents in front of the logarithm.

Understanding these rules helps in maneuvering through logarithmic expressions efficiently. They essentially translate operations from exponential formats into simpler arithmetic ones. Knowing when and how to apply these rules will allow you to tackle even the most difficult problems with confidence and clarity.