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Chapter 3: Problem 37

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

Short Answer

Expert verified
\text{ln} 0.0182 \approx -4.008490

Step by step solution

01

Understand the Natural Logarithm

The natural logarithm, denoted as \(\text{ln}\), is the logarithm to the base \(e\) where \(e\) is an irrational constant approximately equal to 2.71828.
02

Use a Calculator

To find \( \text{ln} 0.0182 \), use a scientific calculator. Utilize the natural logarithm function (often labeled as \( \text{ln} \) or \( \text{LN} \)).
03

Input the Value

Enter the value 0.0182 into the calculator and press the \( \text{ln} \) button.
04

Read the Result

The calculator will display the result. Make sure to round the answer to six decimal places.
05

Verify the Rounding

Double-check the rounding to ensure the result is accurately rounded to six decimal places.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Base e
The natural logarithm is represented by \(\text{ln}\) and is the logarithm to the base \(\text{e}\). The mathematical constant \(\text{e}\) is an irrational number approximately equal to 2.71828. In simpler terms, natural logarithms answer the question: 'To what power must \(\text{e}\) be raised, to get a certain number?' This is different from common logarithms that use base 10. For example, to find \(\text{ln} 0.0182\), we are looking for the power to which \(\text{e}\) must be raised to yield 0.0182. This is especially useful in various fields including natural sciences and economics.
Scientific Calculator
A scientific calculator is highly beneficial for computing natural logarithms. To calculate \(\text{ln} 0.0182\), follow these simple steps:
  • Turn on your calculator and ensure it is set to normal mode, not scientific notation mode.
  • Locate and press the \(\text{ln}\) button, which is dedicated for natural logarithms (This button might also be labeled as LN on some models).
  • Enter the number you need to take the natural logarithm of, in this case, 0.0182.
  • Press the equals button (=) or the enter key.

The calculator will show the result, which needs to be rounded to six decimal places as required.
Rounding
Rounding numbers means reducing the digits of the number while keeping it close to its original value. For this exercise, we need to round the result of \(\text{ln} 0.0182\) to six decimal places. Here's how:
  • Count six digits to the right of the decimal point.
  • If the digit after the sixth place is 5 or greater, increase the sixth digit by one.
  • If the digit after the sixth place is less than 5, keep the sixth digit as is.

For example, if the calculator displays -4.008049345, you would round it to -4.008049 because the seventh digit (4) is less than 5. Always double-check your rounding to avoid small mistakes that could affect your answer.

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