Problem 52 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The exponential form of \(\ln 1=0\) is \(e^0 = 1\)

Step 1: Identify variables from the logarithmic equation

From the equation \(\ln 1=0\) we can identify that our base \(b = e\) (natural logarithm), the argument \(a = 1\), and result \(c = 0\).

Step 2: Apply the fundamental identity of logarithms

The fundamental identity of logarithms states that if \(\log_b a = c\), then \(b^c = a\). Applying this identity to our equation gives us \(e^0 = 1\).

Step 3: Simplify the exponential equation

Since any number (except 0) raised to the power of 0 equals 1, we have our final equation \(e^0 = 1\).

Key Concepts

Logarithmic Equation

A logarithmic equation is an equation that involves a logarithm with an unknown variable. It's helpful to think of logarithms as the inverse of exponentials. When you see an equation like \( \ln 1 = 0 \), it represents the natural log of a number or expression equaling zero.
This equation has a hidden structure that can be revealed by using the properties of logarithms. Here, \(n\) represents the solution to a logarithm which typically answers the question: what exponent of the base yields the given number? In this expression:

• \( \ln(1)\) means "What power must \(e\) be raised to, to equal 1?"
• The solution here is zero because \(e^0 = 1\).

Understanding how to convert logarithmic form to exponential form is important because it simplifies solving equations and solving other related mathematical problems.

Fundamental Identity of Logarithms

The fundamental identity of logarithms is a crucial tool to change logarithmic equations into exponential ones. It states that if \( \log_b a = c \), then \( b^c = a \). This relationship allows us to translate back and forth between logarithmic and exponential forms.
In our example, \( \ln 1=0 \), where the natural logarithm base is \(e\), we apply this identity:

• Identify the base, which in the natural logarithm is \(e\).
• Recognize that \(a = 1\) and \(c = 0\).
• Then rewrite the expression as \(e^0 = 1\).

This method is invaluable when you need to solve or simplify equations, as it often results in simpler calculations and evaluations.

Natural Logarithm

The natural logarithm is a specific logarithm where the base is the mathematical constant \(e\), approximately equal to 2.718. It's denoted by \( \ln\) instead of \( \log_e\), to emphasize its specific nature and frequent use in calculus and growth processes.
The natural logarithm rules and properties are particularly advantageous in science and engineering fields:

• It simplifies complex growth patterns.
• Solves equations involving exponential growth and decay.
• Converts multiplicative processes into additive ones.

In the exercise, understanding \( \ln 1 = 0 \) highlights that any number or process that can reach 1 through multiplication, when base \(e\) is raised to \(0\), will result in 1. This fundamental concept of logarithms is vital for higher-level mathematics and its applications across various domains.