Problem 20 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to expand the expression. $$\log _{3}(5 y)$$

Step-by-Step Solution

Answer

\( \log_{3}(5) + \log_{3}(y) \)

Step 1: Recognize the Logarithmic Product Rule

The expression \( \log_{3}(5y) \) involves the product of 5 and \( y \). The logarithmic product rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. That is, \( \log_{b}(MN) = \log_{b}(M) + \log_{b}(N) \). This is the rule we will use to break down the expression.

Step 2: Apply the Product Rule

Using the product rule identified in Step 1, apply it to the expression \( \log_{3}(5y) \). We split the logarithm into the sum of two logarithms: \( \log_{3}(5) + \log_{3}(y) \).

Key Concepts

Logarithmic Product Rule

The logarithmic product rule is a fundamental concept in logarithms that helps simplify expressions involving products inside a log. When you come across an expression like \( \log_{b}(MN) \), this rule allows you to express it as the sum of two separate logarithms: \( \log_{b}(M) + \log_{b}(N) \). This is extremely useful for breaking down complex expressions into simpler parts.

For example, if you have \( \log_{3}(5y) \), recognize that 5 and \( y \) are being multiplied. According to the product rule, this expression can be split into \( \log_{3}(5) + \log_{3}(y) \). By doing this, you transform a single logarithmic expression into a sum, making calculations or further transformations much easier.

This rule is part of the bigger set of laws governing logarithms, which are designed to manipulate and simplify logarithmic expressions efficiently. It is essential to master this rule as it lays a strong foundation for tackling more complex logarithmic problems.

Logarithmic Expressions

A logarithmic expression involves a logarithm, which is the inverse operation of exponentiation. Understanding how these expressions work is key to solving problems that involve them. They often require you to manipulate them using rules such as the product rule.

Logarithms operate based on certain principles: if \( b^c = a \), then \( \log_{b}(a) = c \). This relationship defines the power \( c \) as the logarithm base \( b \) of \( a \).

In order to work with logarithmic expressions like \( \log_{3}(5y) \), you need to identify whether the expression is a product, quotient, or power. Only then can you effectively apply the appropriate logarithmic rule to expand or simplify the expression.

This knowledge is particularly useful in algebra and calculus, where logarithms are used to solve equations involving exponential growth and decay, compounding interest, and various scientific computations.

Expanding Logarithms

Expanding logarithms involves taking an expression within a logarithm and breaking it down into a sum, difference, or multiple of simpler logs. This process is invaluable in many mathematical scenarios, helping to simplify complex log equations.

In our example, we started with \( \log_{3}(5y) \) and used the product rule to expand it into \( \log_{3}(5) + \log_{3}(y) \). Here, the single logarithmic expression is transformed into a more manageable format.

Expanding logarithmic expressions isn't just about applying rules but also about understanding the nature of the components involved. If the expansion involves powers, you'd use another logarithm rule known as the power rule: \( \log_{b}(M^n) = n \log_{b}(M) \). If it involves division, you'd use the quotient rule.

Mastering these expansions will make you quicker at solving logarithmic equations and help deepen your understanding of how logarithms relate to other mathematical operations.