Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ e $$

Step-by-Step Solution

Answer

\(\ln(e) = 1\)

Step 1: Recall the Definition of Natural Logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant \(e\). It is commonly denoted as \(\ln(x)\).

Step 2: Recognize the Expression

In this problem, you're asked to find \(\ln(e)\). This is the natural logarithm of \(e\) itself.

Step 3: Apply the Property of Logarithms

One important property of logarithms is that \(\ln(e) = 1\) because the natural logarithm of a base lifted to the first power is always 1, or \(b^1 = b\).

Step 4: Round the Result

Since \(\ln(e) = 1\), there is no need for rounding because 1 is already an exact number with no decimal points to round.

Key Concepts

Understanding Logarithmic Properties

Logarithmic properties are crucial for solving problems involving logarithms, like finding the natural logarithm of numbers. Logarithms help us express very large or very small numbers more conveniently. One fundamental property is the relationship between a logarithm and its base. For example, the natural logarithm \(\ln(x)\) is written in terms of base \(e\), which is approximately 2.71828. **Key Properties:**

• \(\log_b(b) = 1\) for any base \(b\). This property indicates that the logarithm of a base to its own power is 1.
• \(\log_b(1) = 0\) since any number raised to the power of zero equals 1.
• The change of base formula helps in converting logarithms from one base to another: \(\log_b(a) = \frac{\log_k(a)}{\log_k(b)}\) for any positive base \(k\).

In our original exercise, knowing that \(\ln(e) = 1\) comes from these properties. Understanding and applying these logarithmic rules simplify many math problems.

Decoding Base e

The mystical base \(e\) surfaces often in mathematical calculations, especially in fields like calculus and complex analysis. But what exactly is \(e\)? It is an irrational number, meaning it cannot be precisely expressed as a fraction, much like \(\pi\). In decimal form, \(e\) starts as 2.71828 and continues indefinitely without repeating.
Base \(e\) is the foundation of natural logarithms, which evolve from processes involving constant growth rates, like population growth or radioactive decay. **Why \(e\) is Special:**

• Exponential processes: Many scientific phenomena can be described by expressions of the form \(e^{x}\), showing growth or decay.
• Compounding interest: In finance, calculating continuous compound interest involves powers of \(e\).

Its ubiquitous nature and applications make \(e\) indispensable in mathematics, and knowing this helps contextualize why natural logarithms are based on \(e\).

Mastering Rounding

Rounding is a mathematical process used to simplify numbers, making them easier to work with, while maintaining their essence. It’s particularly useful in narrowing down numbers to a specific degree of precision, like rounding to the nearest hundredth. Here's how:

• Determine which place value to round to.
• Look at the digit immediately to the right.
• If it's 5 or greater, round up. Otherwise, round down.

In the context of our exercise, finding \(\ln(e)\) resulted in 1. The process required rounding to the nearest hundredth, but since 1 is a whole number already, no rounding was necessary. Understanding how and when to round numbers ensures accuracy, particularly in calculations involving decimals.