Problem 22 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to expand the expression. $$\log _{5} \frac{x}{2}$$

Step-by-Step Solution

Answer

The expanded form is \( \log_{5}(x) - \log_{5}(2) \).

Step 1: Understand the logarithm function

The expression given is a logarithm of a fraction, \( \log_{5}\left(\frac{x}{2}\right) \). The task is to expand it using the laws of logarithms. In this case, we will use the law that deals with division within a logarithmic expression.

Step 2: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \). Applying this to \( \log_{5}\left(\frac{x}{2}\right) \), we get \( \log_{5}(x) - \log_{5}(2) \).

Key Concepts

Laws of Logarithms

Logarithms operate under a set of powerful rules that simplify complex expressions, known as the "laws of logarithms." These laws allow us to manipulate logarithmic expressions, making them easier to solve or interpret. The three main laws include:

• Product Rule: This rule states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, it's expressed as \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \).
• Quotient Rule: This involves division and states that the logarithm of a quotient is the difference between the logarithm of the numerator and the denominator. It's written as \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \).
• Power Rule: This rule handles exponents inside a log and simplifies them by bringing the exponent out as a multiplier: \( \log_b(M^k) = k \cdot \log_b(M) \).

These laws are essential in transforming logarithmic expressions into more manageable forms, especially when solving equations or expanding expressions.

Quotient Rule

The quotient rule is specifically used when dealing with the division of numbers inside a logarithm. In the given problem, \( \log_{5}\left( \frac{x}{2} \right) \), this rule becomes very handy. This rule transforms the division under the log sign into a subtraction operation of two logs.When using the quotient rule, think of the expression as essentially "splitting" the division into subtraction, reducing the complexity:

• The original expression is \( \log_{5}\left( x \right) \) minus \( \log_{5}\left( 2\right) \).

This rule is particularly beneficial when you're trying to break down a log expression into its individual components or when isolating variables in equations.

Logarithmic Expansion

Logarithmic expansion involves breaking down a complex logarithmic expression into simpler terms, using the laws of logarithms. For our exercise, the goal was to expand \( \log_{5}\left( \frac{x}{2} \right) \).Here's how it goes:

• Start with identifying which log law applies: In this case, the Quotient Rule is ideal for expanding \( \log_{5}\left( \frac{x}{2} \right) \).
• Apply this rule to get \( \log_{5}(x) - \log_{5}(2) \). Now, the expression is simplified into two separate logs, making it easier to handle and understand.

Expansion like this is crucial in algebra and calculus because it allows for the separation of variables and individual term investigation. It forms a foundation for more advanced manipulations and solutions of logarithmic equations.