Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. \(\ln (x)=2\)

Step-by-Step Solution

Answer

\( x = e^2 \)

Step 1: Identify the form of the logarithmic equation

The given equation is in the natural logarithm form: \( \ln(x) = 2 \), where \( \ln \) denotes the natural logarithm, which is the logarithm to the base \( e \).

Step 2: Convert logarithmic equation to exponential form

To convert \( \ln(x) = 2 \) into its exponential form, we use the property that \( \ln(x) = 2 \) is equivalent to \( e^2 = x \) because the natural logarithm has a base of \( e \).

Step 3: Solve for x

Now that we have converted the equation to its exponential form \( e^2 = x \), it implies that \( x = e^2 \). Compute this expression to find the value of \( x \).

Key Concepts

Natural Logarithm

The natural logarithm, often represented as \( \ln \), is a special type of logarithm where the base is the mathematical constant \( e \). This constant \( e \) is approximately equal to 2.71828 and is fundamental in many areas of mathematics and science. The natural logarithm helps to model continuous growth processes such as population growth or radioactive decay.

When you encounter \( \ln(x) \), it means "the power to which \( e \) must be raised to obtain \( x \)." For instance, if \( \ln(x) = 2 \), it indicates that \( e \) raised to the power of 2 gives you \( x \). Understanding this concept is essential for converting and solving equations that involve natural logarithms.

Logarithmic to Exponential Conversion

One of the key steps in solving logarithmic equations is understanding how to convert them into exponential form. This involves using the properties of logarithms and exponents. If you have an equation \( \ln(x) = n \), the conversion process entails rewriting this as \( e^n = x \).

• Identify that \( \ln(x) \) means the logarithm with a base \( e \).
• Recognize the equivalence in conversion: logarithmic form to exponential form.

This conversion is crucial because the exponential form is often easier to solve directly. In the context of the original exercise, converting \( \ln(x) = 2 \) to \( e^2 = x \) is a straightforward representation that allows for easier computation and understanding.

Solving for Variables

Solving equations, especially those involving logarithms, often requires isolating the variable of interest. In logarithmic equations, once converted to exponential form, it becomes much simpler to solve for the variable. Let's consider the equation \( e^n = x \).

• After conversion, focus on isolating \( x \).
• Identify the numerical value or expression equivalent to the variable.

In the solved exercise, after converting \( \ln(x) = 2 \) to \( e^2 = x \), solving for \( x \) is immediate. The simplicity of solving for the variable post-conversion is one of the powerful aspects of working with exponential forms.

Base e Logarithms

Base \( e \) logarithms, known as natural logarithms, are integral to calculus and mathematical modeling. The base \( e \) itself arises naturally in scenarios of continuous growth, compounding interest, or decay processes. It's defined as the unique number that makes the statement \( \ln(e) = 1 \).

When working with logarithmic equations:

• Recognize that \( \ln(x) = y \) implies \( e^y = x \).
• The base \( e \) plays a critical role in converting logs to exponential form.

This understanding is crucial, as base \( e \) logarithms frequently appear in calculus and differential equations, providing a bridge between algebra and higher mathematics. In the solved exercise, using \( e^2 \) as a solution illustrates the seamless integration of base \( e \) logarithms in problem-solving.