Problem 15 - College Algebra

Question

For the following exercises, rewrite each equation in exponential form. $$\ln (w)=n$$

Step-by-Step Solution

Answer

The equation in exponential form is \( w = e^n \).

Step 1: Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to base \( e \), where \( e \approx 2.71828 \). Therefore, \( \ln (w) = n \) means that \( w \) is the result of the number \( e \) raised to the power \( n \).

Step 2: Rewriting in Exponential Form

To convert the logarithmic equation \( \ln (w) = n \) into its exponential form, we use the property: a natural logarithm \( \ln(a) = b \) implies \( a = e^b \). Applying this property here, we rewrite \( \ln (w) = n \) as \( w = e^n \).

Key Concepts

Natural Logarithm

The natural logarithm is a special type of logarithm. It uses the base \( e \), which is an irrational constant approximately equal to 2.71828. The natural logarithm is usually represented by \( \ln \). Understanding natural logarithms is crucial, especially if you are dealing with growth processes like populations or interest, as they naturally occur in these contexts.

• Natural logarithms help to simplify equations that involve exponential growth or decay.
• When you see \( \ln(x) \), it's asking: "To what power must \( e \) be raised to equal \( x \)?"

The property of the natural logarithm as described in this exercise is essential: \( \ln(w) = n \) implies that \( e \) raised to the power \( n \) equals \( w \). This lays the foundation to rewrite logarithmic expressions in their exponential forms.

Base e

The number \( e \) plays a fundamental role in mathematics, particularly in calculus and complex numbers. Referred to as the base of the natural logarithm, it is an irrational number. In simpler terms, \( e \) cannot be expressed as a simple fraction.

The significance of \( e \) can be seen in various applications like calculating compound interest, analyzing population growth, and solving differential equations.

• \( e \) is the base for the natural logarithm, forming the inverse relationship in the function \( \ln(x) = y \rightarrow e^y = x \).
• It provides a smooth curve when plotting exponential functions due to its natural properties.

Understanding \( e \) is vital when dealing with exponential equations and logarithms. In this exercise, it helps to convert \( \ln(w) = n \) into exponential form, \( w = e^n \). This transformation is key to solving many logarithmic and exponential problems.

Logarithmic Equation

Logarithmic equations are equations that involve logarithms, which can often be expressed in exponential form. Being comfortable with this conversion is a powerful tool in problem-solving.

A logarithmic equation like \( \ln(w) = n \) states the power to which the base \( e \) must be raised to yield \( w \). Here’s how you can interpret and convert similar equations:

• Recognize the logarithmic form \( \ln(a) = b \).
• Switch it to the exponential form using the power of \( e \): \( a = e^b \).

Converting equations into exponential form makes it easier to comprehend their meaning and apply them to real-world scenarios. In the provided exercise, rewriting \( \ln(w) = n \) as \( w = e^n \) simplifies the equation and clarifies that \( w \) is \( e \) raised to the power \( n \). Practicing this conversion will enhance your skills in tackling logarithmic and exponential expressions.