Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the natural logarithm of each number to the nearest hundredth. $$ 0.342 $$

Step-by-Step Solution

Answer

The natural logarithm of 0.342, rounded to the nearest hundredth, is approximately -1.07.

Step 1: Understanding Natural Logarithms

The natural logarithm of a number is the power to which Euler's number (approximately 2.718) must be raised to yield that number. It is represented as \( \ln(x) \). For this exercise, we will find \( \ln(0.342) \).

Step 2: Use a Calculator

Enter the number 0.342 into a calculator and use the natural logarithm function, often labeled as "ln". The calculator will provide you the value of \( \ln(0.342) \).

Step 3: Round to the Nearest Hundredth

The calculator gives us a value of approximately -1.07359 for \( \ln(0.342) \). We need to round this value to the nearest hundredth. Look at the digit in the thousandth place: if it's 5 or more, round up the hundredth place. Otherwise, keep the hundredth place the same.

Key Concepts

Euler's Number

Euler's number, represented by the letter \( e \), is a mathematical constant that is approximately equal to 2.718. This irrational number plays a crucial role in various areas of mathematics, especially in calculus and complex analysis. It is named after the Swiss mathematician Leonhard Euler.
In the context of natural logarithms, \( e \) is the base of the natural logarithm function. The natural logarithm of a number \( x \), denoted as \( \ln(x) \), provides the power to which \( e \) must be raised to yield \( x \).
For example:

• \( \ln(e) = 1 \) because \( e^1 = e \).
• \( \ln(1) = 0 \) since \( e^0 = 1 \).

The relationship between exponentials and logarithms is a fundamental concept, and understanding \( e \) is essential for solving problems involving natural logarithms.

Rounding Decimals

Rounding is the process of simplifying a number while keeping its value close to what it was. In mathematics, rounding to the nearest hundredth means adjusting the number to two decimal places. This technique is essential when dealing with long decimal results, making them easier to understand or compare.
Here's how you round to the nearest hundredth:

• Identify the digit in the hundredth place (the second digit after the decimal point).
• Look at the digit in the thousandth place (the third digit after the decimal point).
• If the thousandth digit is 5 or greater, increase the hundredth digit by one.
• If it is less than 5, leave the hundredth digit unchanged.

For instance, the value \(-1.07359\) becomes \(-1.07\) after rounding to the nearest hundredth because the thousandth digit is 5.

Using a Calculator

Calculators are indispensable tools in modern mathematics, especially when dealing with complex operations like finding natural logarithms. They help in performing tedious calculations quickly and accurately, allowing us to focus on understanding the core concepts.
To use a calculator for natural logarithms:

• Ensure your calculator has the "ln" function, which represents the natural logarithm.
• Enter the number whose logarithm you wish to calculate, such as 0.342.
• Press the "ln" button to get the result, which in this case will be approximately \(-1.07359\).

By efficiently using calculators, students can easily find logarithmic values and then proceed to apply or interpret these values in various contexts, such as solving exponential equations or analyzing growth models.