Problem 126 - Introductory and Intermediate Algebra for College Students 4th

Question

$$\text { Evaluate } 3 \ln (2 x) \text { if } x=\frac{e^{4}}{2}$$

Step-by-Step Solution

Answer

The result is 12.

Step 1: Substitution

Replace \(x\) in the given expression with \(\frac{e^{4}}{2}\), thus the expression becomes \(3 \ln (2(\frac{e^{4}}{2}))\).

Step 2: Simplify Inside the Logarithm

Simplify the expression inside the logarithm to validate: \(3 \ln (e^{4})\).

Step 3: Apply the Logarithm Property

Recall the property of the natural logarithm: \(\ln(a^b) = b \ln(a)\). Thus \(3 \ln (e^{4})\) can be rewritten as \(3 \cdot 4\).

Step 4: Evaluate the Expression

Evaluate \(3 \cdot 4\) to get the final result.

Key Concepts

Natural Logarithm

The natural logarithm, often written as \( \ln \), is a special type of logarithm with a base of \( e \). \( e \) is approximately equal to 2.71828, and is a very important constant in mathematics. Natural logarithms are used frequently in calculus and in solving exponential growth problems. Unlike common logarithms, which have a base of 10, natural logarithms deal with continuous growth rates and are common in scientific calculations.

In this exercise, the expression involves the natural logarithm of \( 2x \). The natural logarithm simplifies calculations when the base is \( e \), because one of the key features is that \( \ln(e^x) = x \). Therefore, when evaluating \( \ln(e^4) \), as seen in the solution, it simplifies directly to 4.

Logarithm Properties

Understanding logarithm properties is essential for simplifying expressions, especially in problems involving natural logarithms. Here are some fundamental properties of logarithms to remember:

• The product property: \( \log_b(xy) = \log_b x + \log_b y \)
• The quotient property: \( \log_b(\frac{x}{y}) = \log_b x - \log_b y \)
• The power property: \( \log_b(x^a) = a \log_b x \)

In the solution step provided, the property \( \ln(a^b) = b\ln(a) \) was used. This property allows you to move the exponent out front, simplifying the computation. Therefore, \( \ln(e^4) \) simplifies to \( 4 \cdot \ln(e) \), which equals 4 because \( \ln(e) = 1 \). Knowing these properties can make evaluating logarithmic expressions much easier.

Substitution Method

The substitution method is a straightforward technique used to replace variables with given values. This method is particularly useful in simplifying expressions and solving algebraic equations.

In the original problem, you need to evaluate \( 3 \ln(2x) \) with \( x = \frac{e^4}{2} \). The first step in solving this exercise was to substitute \( x \) with \( \frac{e^4}{2} \). This makes the expression \( 3 \ln(2(\frac{e^4}{2})) \).

By carrying out the multiplication inside the parentheses, the expression simplifies to \( 3 \ln(e^4) \). The use of the substitution method allows you to tackle complex expressions in a systematic way by replacing variables with numeric values to simplify the calculation and proceed with evaluation.