Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ e^{2} $$

Step-by-Step Solution

Answer

The natural logarithm of \( e^2 \) is 2.00.

Step 1: Understand the Problem

We need to find the natural logarithm of a given number, which in this case is \( e^2 \), and round the result to the nearest hundredth.

Step 2: Recall Logarithm Properties

The natural logarithm (\( \ln \)) of a number that is an exponent of \( e \), such as \( e^x \), is simply the exponent itself. Thus, \( \ln(e^{x}) = x \).

Step 3: Apply the Logarithm Rule

Since our expression is \( e^2 \), applying the property \( \ln(e^{x}) = x \), we find that \( \ln(e^2) = 2 \).

Step 4: Round to the Nearest Hundredth

The value obtained, 2, is already a whole number. Therefore, its representation rounded to the nearest hundredth is 2.00.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent tells you how many times to multiply the base by itself.
For example:

• For the expression \( e^2 \), the base is \( e \) (approximately 2.718), and the exponent is 2, which means you multiply \( e \) by itself.
• This results in \( e \times e \), giving us a concise way to represent repeated multiplication.

The process of exponentiation is crucial for understanding natural logarithms and many other mathematical concepts. It provides a foundational understanding needed to delve into logarithms, where exponentiation operates in reverse.

Logarithm Properties

Logarithms are all about determining what exponent you put on a base to obtain a certain number. The natural logarithm, denoted as \( \ln \), uses the base \( e \). One of the most important properties of logarithms is the relationship with exponentiation:

• If you have \( e^x \), then the natural logarithm of \( e^x \) is simply \( x \).
• This property is expressed as \( \ln(e^x) = x \), effectively cancelling out the exponential and leaving you with the exponent.

This property is highlighted by our problem: finding \( \ln(e^2) \). Without needing to perform any complex calculations, the natural logarithm property helps us recognize that the result is 2.

Mathematical Rounding

Rounding is a mathematical technique used to make numbers simpler and to keep them within a certain level of precision. When rounding to the nearest hundredth, you look at the number in the thousandths position:

• If that number is 5 or greater, you round the number in the hundredths position up.
• If it's less than 5, you leave the hundredths digit unchanged.

In our specific case of \( \ln(e^2) = 2 \), the result is a whole number. Therefore, when rounding to the nearest hundredth, we express it as 2.00. This notation indicates precision to two decimal places, aligning with rounding requirements even when the additional digits are zero.