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Chapter 12: Problem 20

Find each logarithm. Give approximations to four decimal places. \(\ln 942.6\)

Short Answer

Expert verified
6.8494

Step by step solution

01

- Understand the notation

The notation \(\backslash\text{ln}\) represents the natural logarithm, which is the logarithm to the base e (where e is approximately 2.71828).
02

- Use a calculator

To find \(\backslash\text{ln}(942.6)\), input 942.6 into a scientific calculator and press the \(\backslashtext{ln}\) function.
03

- Approximate the value

The calculator will provide the value. Ensure to round the result to four decimal places.

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Key Concepts

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

Understanding Logarithms
Logarithms are mathematical functions that help us understand exponents. If you鈥檙e given a number and want to know what power you must raise a specific base to get that number, you use a logarithm. The natural logarithm, denoted as \(\text{ln}\), uses the base \(e\), which is an irrational number approximately equal to 2.71828.

Essentially, \(\text{ln}(x)\) answers the question: 'To what power must \(e\) be raised to produce \(x\)?'. For instance, \(\text{ln}(e) = 1\) because \(e\) raised to the power of 1 is \(e\). It鈥檚 a crucial concept in many areas such as calculus, complex numbers, and even real-world applications like compound interest and population growth.
Using a Scientific Calculator
A scientific calculator can perform various complex calculations, including finding logarithms. Here鈥檚 a general guide on how to use it for natural logarithms:

  • Turn on your calculator and ensure it鈥檚 in the normal operation mode (not in a programming or statistical mode).
  • Locate the \(\text{ln}\) button, which stands for natural logarithm.
  • Enter the number for which you want to find the logarithm. For example, to find \(\text{ln}(942.6)\), type 942.6.
  • Press the \(\text{ln}\) button after typing in the number. The calculator will compute and display the natural logarithm of the entered number.


With practice, using these functions will become second nature, aiding in efficient and accurate mathematical computations.
Rounding to Four Decimal Places
Rounding helps simplify numbers to make them easier to work with, while still being quite accurate. When asked to round to four decimal places, you should:

  • Look at the fifth decimal place.
  • If the fifth digit is 5 or greater, round the fourth decimal place up by one.
  • If the fifth digit is less than 5, leave the fourth decimal place as it is.


For example, if we鈥檙e rounding the result of \(\text{ln}(942.6)\), which is approximately 6.8499:

  • Inspect the fifth decimal place (in this case, it might be 8 if the real value was 6.84998).
  • Since 8 is greater than 5, we round up the fourth decimal place from 9 to 0 (carrying over to make the number 6.8500).

This ensures your results are precise and standardized, making them easier to compare and analyze.

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Most popular questions from this chapter

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places. \(\log _{6} \sqrt[3]{5}\)

Use your calculator to find approximations of the following logarithms. (a) \(\log 356.8\) (b) \(\log 35.68\) (c) \(\log 3.568\) (d) Observe your answers and make a conjecture concerning the decimal values of the common logarithms of numbers greater than 1 that have the same digits.

Graph each exponential function. $$ g(x)=\left(\frac{1}{5}\right)^{x} $$

Find the hydronium ion concentration of the substance with the given pH. Spinach, 5.4

Let \(f(x)=2^{x} .\) We will see in the next section that this function is one- toone. Find each value, always working part (a) before part \((b)\). (a) \(f(-2)\) (b) \(f^{-1}\left(\frac{1}{4}\right)\)
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