Problem 45 - Calculus and its applications

Question

Find each logarithm. Round to six decimal places. $$ \ln 0.0182 $$

Step-by-Step Solution

Answer

The natural logarithm of 0.0182 is approximately -4.005333.

Step 1: Understand the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \approx 2.7182818 \). The value \( \ln(0.0182) \) represents the power to which we must raise \( e \) to obtain 0.0182.

Step 2: Use a Calculator

Since \( 0.0182 \) is less than 1, its natural logarithm will be a negative number. Use a scientific calculator to find \( \ln(0.0182) \). Input 0.0182 and press the natural log (\( \ln \)) button.

Step 3: Record and Round the Result

The calculator will display the result as \( -4.005332881 \). Round this result to six decimal places to get \( -4.005333 \).

Key Concepts

logarithms

Logarithms are a mathematical concept that helps us understand the operation of exponentiation in reverse. The logarithm of a number is what power we need to raise a certain base to get that number. The base can vary:

• Common logarithms use base 10 and are often used in scientific calculations.
• Natural logarithms use Euler's number, approximately 2.7182818, as the base.

Logarithms help to simplify complex calculations, like multiplication and division, into simpler ones, like addition and subtraction.
When dealing with natural logarithms, denoted as \(\ln(x)\), you are finding out the exponent that Euler's number \(e\) must be raised to, in order to get \(x\). For instance, if you have \(\ln(0.0182)\), it means you're finding the exponent for which \(e^{\text{exponent}} = 0.0182\). This exponent will be a negative number because \(0.0182 < 1\).

euler's number

Euler’s number, denoted as \(e\), is a fundamental constant in mathematics, similar to \(\pi\). It’s approximately equal to 2.7182818 and is the base for natural logarithms.
Why is it so important?

• It appears in growth processes, like compound interest and population growth.
• It's a critical component in calculus, particularly in finding the derivative and integral of functions involving exponentials.
• It relates complex numbers with trigonometric functions through Euler's formula, \(e^{ix} = \cos(x) + i\sin(x)\).

This number connects various mathematical concepts, making it a key element in mathematical analysis and applications. Thus, knowing how to find the natural logarithm essentially means understanding the relationship between \(x\) and \(e\) in the context \(e^{f(x)} = x\).

rounding decimals

Rounding decimals is a crucial skill, especially in scientific and mathematical contexts where precision matters. When instructed to round to a certain number of decimal places, you need to follow specific steps:

• Identify the decimal place you're rounding to. For example, when rounding to six decimal places, you focus on the sixth digit after the decimal point.
• Look at the digit immediately following it. If this digit is 5 or greater, round up the last number you're keeping. If it's less than 5, keep the number as is.

For \(\ln(0.0182) = -4.005332881\), we round to six decimal places. The seventh digit (8) requires us to increase the sixth decimal place by one. Thus, the rounded result becomes \(-4.005333\).
Rounding helps manage the level of precision in calculations, ensuring our results reflect the desired accuracy levels, especially when using a calculator or computer that outputs many more digits than necessary.