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Chapter 12: Problem 20

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

Short Answer

Expert verified
at fraction {.3333, .6667 \text The natural logarithm of 942.6 is approximately 6.8491.

Step by step solution

01

Understand the task

The exercise asks to find the natural logarithm (base e) of the number 942.6 and round the answer to four decimal places.
02

Use a calculator

Enter the value 942.6 into a scientific calculator and then apply the natural logarithm function (often labeled as 'ln' on the calculator). This function will automatically compute the natural logarithm of the input number.
03

Approximate to four decimal places

Observe the result displayed by the calculator which should be a long decimal number. Round this number to four decimal places carefully.

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Key Concepts

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

logarithm
In mathematics, a logarithm is the opposite of exponentiation. While an exponentiation involves raising a number to a power, a logarithm does just the reverse 鈥� it tells us what power we need to raise a given base to obtain another number. For example, in the equation \( \text{log}_b(a) = c \), b raised to the power of c equals a. This concept is essential in both algebra and calculus.

Logarithms can have different bases. The most common bases are 10 (common logarithm) and 'e' (natural logarithm). Each base has its own unique properties and uses. Making sure you understand the base is crucial for solving logarithmic functions accurately.
natural logarithm
A natural logarithm is a logarithm to the base 'e', where 'e' is an irrational number approximately equal to 2.71828.

It's often abbreviated as 'ln'. For example, the natural logarithm of a number x is written as \( \text{ln}(x) \). The natural logarithm function plays a significant role in calculus, particularly in dealing with growth and decay problems.

To compute a natural logarithm:
  • Use a scientific calculator.
  • Input the number (in our case, 942.6).
  • Press the 'ln' button to get the result.

The answer will be a decimal that you may need to round, as shown in the exercise.
rounding decimals
Rounding decimals is the process of trimming a decimal number to a specific place value for simplicity or to meet particular requirements. For example, rounding to four decimal places means adjusting the number so that only four digits come after the decimal point.

To round a number correctly:
  • Identify the digit in the place value you are rounding to (fourth decimal place here).
  • Check the digit immediately to the right (the fifth decimal place).
  • If this digit is 5 or higher, increase the fourth place value by 1. If it's 4 or lower, keep the fourth place value as is.

This rounding method ensures consistency and accuracy. For example, a result like 6.85057 becomes 6.8506 after rounding to four decimal places.
scientific calculator usage
Scientific calculators are powerful tools designed to handle complex mathematical functions with ease. Using them wisely can save much time and effort especially when solving logarithmic problems. To use a scientific calculator for computing a natural logarithm:

  • Turn on the calculator.
  • Enter the number 942.6.
  • Locate and press the 'ln' button to compute the natural logarithm.

The calculator will display the result, which may be a long decimal number. If required, round the number to the necessary decimal places, as discussed earlier.

Always make sure your calculator is in the correct mode for the calculations you are performing, and verify your entries before proceeding with operations.

These steps will ensure accurate and efficient use of your calculator.

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Most popular questions from this chapter

Solve equation. \(\log _{1 / 2} 8=x\)

If the function is one-to-one, find its inverse. \(f(x)=2 x+4\)

Each function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window. \(f(x)=2 x-7\)

If the function is one-to-one, find its inverse. \(\\{(-8,6),(-4,3),(0,6),(5,10)\\}\)

If the function is one-to-one, find its inverse. \(f(x)=3 x+1\)
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