Problem 67 - Algebra 2

Question

Which expression is equal to \(3 \ln 4-5 \ln 2 ?\) $$ \begin{array}{llll}{\text { A. } \ln (-18)} & {\text { B. } \ln \left(\frac{6}{5}\right)} & {\text { C. } \ln 2} & {\text { D. } \ln 32}\end{array} $$

Step-by-Step Solution

Answer

The expression \(3 \ln 4-5 \ln 2\) simplifies to \(ln 2\), therefore the correct choice is C. \(ln 2\).

Step 1: Apply the Property of Logarithms

Firstly, utilize the natural logarithm property \(c*ln(a)=ln(a^c)\) to rewrite the given expression: \(3 \ln 4-5 \ln 2\) is equivalent to \(ln(4^3) - ln(2^5)\). So the new expression becomes \(ln(64) - ln(32)\).

Step 2: Simplify the Expression

Following the simplification, the logarithm property \(ln(a)-ln(b)=ln\left(\frac{a}{b}\right)\) could be utilized to rewrite the expression into a simpler form: \(ln (64) - ln 32\) is equivalent to \(ln \left(\frac{64}{32}\right)\). Therefore, the simplified expression is \(ln 2\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms can greatly simplify many complex mathematical problems. These properties act as rules that allow us to manipulate and transform logarithmic expressions more easily. Let's look at two essential properties used frequently:

• Product Rule: This property states that the logarithm of a product is the sum of the logarithms of the factors. Formally, \( \ln(ab) = \ln a + \ln b \).
• Quotient Rule: This property tells us that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. It is represented as \( \ln(\frac{a}{b}) = \ln a - \ln b \).
• Power Rule: A particularly powerful property, it states that the logarithm of a number raised to a power is the power times the logarithm of the number: \( c \ln a = \ln(a^c) \).

These rules form the foundation for simplifying logarithmic expressions, making them easier to manage. By applying them correctly, complex expressions become much more straightforward.

Natural Logarithms

Natural logarithms are logarithms with the base \( e \), where \( e \approx 2.718 \). This special base appears frequently in mathematics, particularly in calculus, because of its natural growth properties. When we see the notation \( \ln \), we understand it to mean the natural logarithm.Natural logarithms share all the same properties as standard logarithms, such as the product, quotient, and power rules. They have some specific uses:

• Modeling Natural Phenomena: They are used to model phenomena in nature, such as population growth, radioactive decay, and continuous compound interest.
• Calculus Applications: They are particularly convenient in calculus due to their differential and integral properties.

Because of these applications, natural logarithms are a fundamental part of mathematics and sciences, used to solve real-world problems effectively.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using the properties of logarithms to make the expression more compact or easier to compute. Let's break down the simplification process:First, identify and apply any relevant properties of logarithms, such as the power and quotient rules. For instance, given the expression \( 3 \ln 4 - 5 \ln 2 \), we utilized the power rule to rewrite it as \( \ln(4^3) - \ln(2^5) \), simplifying to \( \ln 64 - \ln 32 \).Next, apply the quotient rule to further simplify. Here, \( \ln 64 - \ln 32 \) becomes \( \ln(\frac{64}{32}) \), which simplifies to \( \ln 2 \).Some helpful tips:

• Break it Down: Start from larger expressions and use logarithm rules to break them into simpler parts.
• Check Your Work: Always verify your final simplified expression by ensuring no further properties can be applied.

By following these steps, complex logarithmic expressions can be transformed into a simpler form, making calculations more efficient and comprehensible.