Problem 69 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{5} \frac{x}{25}\)

Step-by-Step Solution

Answer

The expanded form of the logarithm \( \log_{5} \frac{x}{25} \) is \( \log_{5}(x) - 2 \)

Step 1: Apply the Logarithm Division Rule

According to the logarithmic law of division, \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), the given expression \( \log_{5} \frac{x}{25} \) can be written as \( \log_{5}(x) - \log_{5}(25) \).

Step 2: Simplify Constant Logarithm

The constant expression \( \log_{5}(25) \) equals 2 because the base of the logarithm raised to 2 gives 25, simplifying it we get \( \log_{5}(x) - 2 \).

Key Concepts

Logarithm Expansion

Understanding the concept of logarithm expansion involves breaking down complex logarithmic expressions into simpler components. This can be incredibly helpful when you want to work with equations that include products, quotients, or powers inside the logarithm. By expanding a logarithm, you can transform a complicated problem into a series of easier ones.

The basic idea is to use the properties of logarithms to rewrite expressions. For example, the expression \( \log_{b}(mn) \) can be expanded to \( \log_{b}(m) + \log_{b}(n) \). Similarly, an expression involving a quotient, like \( \log_{b}(\frac{m}{n}) \), can be rewritten as \( \log_{b}(m) - \log_{b}(n) \).

It's essential to understand this property because it allows you to deal with logarithmic expressions more flexibly and can provide a straightforward path to solutions when working with logarithmic equations or inequalities. By memorizing and practicing these expansions, you will gain a powerful skill that simplifies numerous algebraic problems.

Logarithmic Division Rule

The logarithmic division rule is a useful property when working with fractions or quotients inside a logarithm. It states that for a logarithm with a fraction as its argument, the expression can be written as the difference of two logarithms. This property stems from the fundamental definition of logarithms as exponents.

For example, the expression \( \log_{b}(\frac{m}{n}) \) can be simplified to \( \log_{b}(m) - \log_{b}(n) \). In the given exercise with the expression \( \log_{5}(\frac{x}{25}) \), we apply this rule:

• The numerator 'x' becomes \( \log_{5}(x) \).
• The denominator '25' becomes \( \log_{5}(25) \).
• This converts the original expression into \( \log_{5}(x) - \log_{5}(25) \).

Understanding and utilizing this rule can simplify many logarithmic expressions, transform them into manageable forms, and make it easier to perform subsequent mathematical operations.

Simplifying Logarithms

Simplifying logarithms is an important step in logarithmic manipulation, essential for making expressions more straightforward and easier to work with. Often, this involves converting logarithms into simpler forms using known values or logarithmic properties.

Take, for instance, the expression \( \log_{5}(25) \) within our solution. Recognizing that 25 is a power of the base (5), specifically \( 5^2 \), allows you to simplify the logarithm. Since \( 5^2 = 25 \), the logarithm \( \log_{5}(25) \) simplifies to 2.

Here's a step-by-step breakdown:

• Recognize that the constant expression inside the logarithm, 25, is \( 5^2 \).
• The logarithm \( \log_{5}(25) \) becomes 2 because you're essentially asking "5 raised to what power equals 25?"
• Therefore, the expression simplifies to \( \log_{5}(x) - 2 \).

By simplifying logarithms, you reduce the complexity of mathematical problems, making them less cumbersome to solve and more intuitive to understand. This process can reveal insights or solutions that might not be immediately apparent.