Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

Which property of logarithms can you use to condense the expression \(\ln x-\ln 2 ?\)

Step-by-Step Solution

Answer

The expression \(\ln x - \ln 2 \) can be condensed to \(\ln(x/2)\) using the quotient property of logarithms.

Step 1: Identifying the Suitable Property

The properties of logarithms include product, quotient, and power properties. Considering the provided expression \(\ln x - \ln 2 \), the quotient property of logarithms is appropriate as this property states that the difference of two logarithms is equivalent to the logarithm of the quotient of their numbers. The quotient rule is given by \(\log_bA - \log_bB = \log_b(A/B)\).

Step 2: Applying the Quotient Property

As such, the expression \(\ln x - \ln 2 \) can be condensed by applying the quotient property. Thus the expression becomes \(\ln(x/2)\).

Key Concepts

Properties of Logarithms

Logarithms have certain properties that make them incredibly useful for simplifying expressions and solving equations. These properties allow us to manipulate expressions in ways that can reveal hidden relationships or simplify complex problems. Here are three core properties of logarithms you should know about:

• Product Property: This property states that the logarithm of a product is equal to the sum of the logarithms of the factors. In mathematical terms, it can be expressed as \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \).
• Quotient Property: This is the property we focus on more in our exercise. It states that the logarithm of a quotient is the difference of the logarithms. In formula form, \( \log_b(A/B) = \log_b(A) - \log_b(B) \).
• Power Property: This property explains that the logarithm of a power is equal to the exponent times the logarithm of the base. Mathematically, \( \log_b(M^n) = n \cdot \log_b(M) \).

These properties help in "condensing" or "expanding" logarithmic expressions and are fundamental for many mathematical concepts involving logarithms.

Quotient Property

The Quotient Property of logarithms is a powerful tool for simplifying expressions. It's crucial to understand this property because it allows you to compress two separate logarithms into one.
When you have the expression \( \log_b A - \log_b B \), this property comes into play by transforming it into a more streamlined form: \( \log_b (A/B) \). This is useful in various mathematical contexts where simplifying expressions can make calculations easier or provide clearer insights.
It reduces the complexity by changing subtraction into division. Thus, when you see a subtraction operation in logarithmic form, think of the quotient property as the go-to tool for consolidation. Remembering this property helps you work quicker and more efficiently with logarithms in algebra and calculus.

Condensing Logarithmic Expressions

Condensing logarithmic expressions refers to the process of combining multiple logarithms into a single logarithmic expression. This process is particularly useful when solving logarithmic equations or simplifying complex logarithmic expressions.
The skills you need to condense logarithms effectively include familiarity with the main properties of logarithms (like those we discussed earlier: the product, quotient, and power properties). You apply these properties to turn lengthy algebraic expressions into concise forms. For example, using the quotient property, you can convert something like \( \ln x - \ln 2 \) into a single condensed expression: \( \ln (x/2) \).
By condensing, you reduce the number of terms, making it easier to analyze and solve the problem at hand. It's an essential skill for math students to develop, as it lays the groundwork for tackling more advanced mathematical problems with confidence and clarity.