Problem 53 - College Algebra

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: Recall the property of logarithms

Recall the fundamental property of logarithms that states \[ \ln(a^b) = b \ln(a) \]for any positive number \( a \) and real number \( b \).

Step 2: Apply the property to simplify

Apply the property to the given expression:\[ 25 \ln \left(e^{\frac{2}{5}}\right) = 25 \times \frac{2}{5} \times \ln(e) \]Since the base of the logarithm and the base of the exponent are the same, we simplify \( \ln(e) = 1 \). Thus, the expression becomes:\[ 25 \times \frac{2}{5} \times 1 \]

Step 3: Perform the arithmetic simplification

Carry out the multiplication:\[ 25 \times \frac{2}{5} = \frac{25 \times 2}{5} = \frac{50}{5} = 10 \].

Key Concepts

Properties of Logarithms

Logarithms have properties that make calculations and simplifications easier. One important property is the power rule. This states that when you have a logarithm like \( \ln(a^b) \), you can simplify it to \( b \ln(a) \). This property allows you to move the exponent in front of the logarithm. Understanding these properties can help you break down complex expressions into simpler forms.

Another key property is the fact that \( \ln(e) = 1 \). This holds true since \( e^1 = e \), the base of the natural logarithm. This simplifies calculations when dealing with expressions containing \( e \).

The laws are particularly useful when simplifying expressions, solving equations, and manipulating mathematical operations in algebra. They can save time and reduce errors when working with logarithmic expressions.

Simplifying Logarithmic Expressions

When it comes to simplifying logarithmic expressions, recognizing when to apply the right properties is crucial. With the expression \( 25 \ln(e^{\frac{2}{5}}) \), the power rule helps break down the terms. Here, noticing that the exponent is \( \frac{2}{5} \), means you can rewrite the expression as \( 25 \times \frac{2}{5} \times \ln(e) \).

Since \( \ln(e) = 1 \), you can remove the logarithm part completely, simply multiplying the remaining numbers. This step-by-step simplification transforms a seemingly complex logarithmic equation into a straightforward arithmetic one.

Remember, always keep an eye on the properties of logarithms as you go through each term. They allow you to rewrite the expression in a more manageable form, thus easing up your calculations.

Arithmetic Simplification

After applying the property of logarithms, what remains is a simple multiplication. In this case, you want to simplify \( 25 \times \frac{2}{5} \). To do this, recall that multiplication involving fractions should follow the "multiply across" rule.

First, multiply the numerator of the fraction by the whole number, which is \( 25 \times 2 = 50 \). Then divide this result by the denominator of the fraction, giving you \( \frac{50}{5} = 10 \).

This arithmetic simplification is often the final step in handling logarithmic expressions. It transforms the expression into an easily understandable numerical value.