Problem 53 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the natural logarithmic expression without using a calculator. \(25 \ln \left(e^{\frac{2}{5}}\right)\)

Step-by-Step Solution

Answer

The expression evaluates to 10.

Step 1: Identify the Logarithmic Property to Use

First, we'll identify the property of logarithms that will help us simplify the expression. We know from logarithmic properties that \( \ln(a^b) = b \cdot \ln(a) \). Here, \( a = e \) and \( b = \frac{2}{5} \). Since the natural logarithm \( \ln \) and the base of the exponential function \( e \) are inverses, \( \ln(e) = 1 \).

Step 2: Apply the Logarithmic Property

Using the property from Step 1, transform the inner part of the expression: \( \ln \left(e^{\frac{2}{5}}\right) = \frac{2}{5} \cdot \ln(e) \). Knowing that \( \ln(e) = 1 \), we simplify to \( \frac{2}{5} \cdot 1 = \frac{2}{5} \).

Step 3: Multiply by the External Factor

Now take the result from Step 2 and multiply it by the factor outside the logarithm: \( 25 \cdot \frac{2}{5} \).

Step 4: Simplify the Expression

Perform the multiplication from Step 3: \( 25 \times \frac{2}{5} = \frac{25 \times 2}{5} = \frac{50}{5} = 10 \).

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools in mathematics that simplify complex expressions. They help us solve logarithmic and exponential equations more easily.
One key property to consider is the Power Rule for logarithms. This property states that a logarithm of a number raised to an exponent can be simplified by multiplying the exponent with the logarithm of the base:

• \( \ln(a^b) = b \cdot \ln(a) \)

In our exercise, we apply this property to \( \ln(e^{\frac{2}{5}}) \). Here, \( a = e \) and \( b = \frac{2}{5} \).
Moreover, a crucial understanding is remembering that the natural logarithm function, \( \ln \), and the exponential function with base \( e \), are inverses. This implies that \( \ln(e) = 1 \). This property is exceptionally useful because it allows expressions involving the natural logarithm of \( e \) to simplify significantly.

Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent.
In simpler terms, it's an operation that involves multiplying a base with itself multiple times to reach a stated power or exponent.

• The natural logarithm \( \ln \) is conventionally used with base \( e \), which is approximately 2.718.

In our scenario, we have the exponential expression \( e^{\frac{2}{5}} \). Here we need to understand two things: the base \( e \), and the fractional exponent \( \frac{2}{5} \).
The fractional exponent indicates a power and a root. Specifically, \( e^{\frac{2}{5}} \) can be conceptualized first as taking the fifth root of \( e^2 \) or as \( \sqrt[5]{e^2} \).
Understanding this relationship helps us with evaluations, especially when logarithms are involved.

Simplifying Expressions

To simplify expressions involving natural logarithms and exponential functions, we utilize properties and identities that we know.
In practical terms, simplifying means transforming a complicated expression into an easier-to-understand or compute form.

• For the original expression \( 25 \ln(e^{\frac{2}{5}}) \), we aim to simplify it step by step.

First, we apply the logarithmic property \( \ln(e^{\frac{2}{5}}) = \frac{2}{5} \) (since \( \ln(e) = 1 \)). Then, our task becomes a simple multiplication by the outside factor 25.
So, we multiply \( 25 \times \frac{2}{5} \) to reach \( \frac{50}{5} \), resulting in the simplified final answer of 10.
This approach demonstrates that through understanding and applying basic mathematical properties, we can simplify seemingly complex expressions effectively.