Problem 253 - Calculus Volume 1

Question

Write the equation in equivalent exponential form. \(\ln 1=0\)

Step-by-Step Solution

Answer

\(e^0 = 1\)

Step 1: Understand the Natural Logarithm

The natural logarithm, \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately 2.718. The expression \(\ln x = y\) means \(e^y = x\). In this problem, \(\ln 1 = 0\).

Step 2: Identify the Base and Result

Since \(\ln\) has a base \(e\), we set \(y = 0\) and \(x = 1\) from the expression \(\ln x = y\). This implies \(e^0 = 1\).

Step 3: Write the Equivalent Exponential Equation

From the equation \(\ln 1 = 0\), write it in exponential form as \(e^0 = 1\). This illustrates that raising \(e\) to the power of 0 results in 1, confirming the equivalence of the forms.

Key Concepts

Exponential Form

The exponential form of an equation is very straightforward once you know the basics. It represents a relationship where a number, known as the base, is raised to a particular power or exponent. The general rule for rewriting a natural logarithm in exponential form is to understand that the expression \( \ln x = y \) can be transformed to the equation \( e^y = x \). Here, \( e \) is the base of the natural logarithm, and \( x \) and \( y \) are the specific values from the logarithmic equation.
For instance, if you have \( \ln 1 = 0 \), you can convert this into its exponential form by recognizing that the base is \( e \), the exponent is 0, and the result, \( x \), is 1. Thus, the equation becomes \( e^0 = 1 \). This tells us that when \( e \) is raised to the power of 0, it equals 1. It’s a simple yet profound representation of how different mathematical forms can convey the same information.

Base e

The number \( e \) is a mathematical constant that is approximately equal to 2.718. It is called Euler's number and is significant in many areas of mathematics, especially in exponential functions and logarithms. This number serves as the base for the natural logarithm.

• It is irrational, meaning it cannot be precisely expressed as a fraction.
• It is transcendental, which implies that it is not a root of any non-zero polynomial equation with rational coefficients.

Using \( e \) as a base in logarithmic functions like \( \ln x \) results in natural logarithms, which are common in calculations involving growth and decay processes, such as population growth, radioactive decay, and compound interest. The base \( e \) brings with it unique properties and benefits, making it an essential tool for solving logarithmic and exponential equations.

Logarithmic and Exponential Relationship

Understanding the link between logarithms and exponents is crucial for mastering these mathematical concepts. The natural logarithm and exponential functions are inverse operations. This means they undo each other when applied in succession.
For example, if you start with a number and take the natural logarithm of it, you can return to the original number by raising \( e \) to the power of the logarithm's result. In symbolic terms, if \( \ln x = y \), then it must be true that \( e^y = x \). This is a critical concept, as it shows the double-sided nature of exponents and logs:

• The logarithm tells you what exponent is needed to produce a given number from a specific base.
• The exponential expression is the actual computation of that number using the base and its exponent.

So, when you transform \( \ln 1 = 0 \) into the exponential form \( e^0 = 1 \), you are simply showing these inverse operations in action, confirming they both express the same relationship in different ways.