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Chapter 10: Problem 12

Find each logarithm. Give approximations to four decimal places. \(\log 0.1741\)

Short Answer

Expert verified
-0.7597

Step by step solution

01

Understand the Logarithm Function

The logarithm function \(\text{log}(x)\) determines the power to which the base, usually 10 in this case, must be raised to obtain the number \(x\). Here, \(x = 0.1741\).
02

Use a Scientific Calculator

To find the logarithm of 0.1741, use a scientific calculator. Ensure the calculator is set to the base 10 logarithm (common logarithm). Input 0.1741 and press the log button.
03

Note the Calculator Result

After performing the calculation, the result displayed on the calculator for \( \text{log}(0.1741) \) should be approximately -0.7597.

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Key Concepts

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

logarithm function
Logarithms can seem tricky at first, but they are an essential part of mathematics. The logarithm function, represented as \(\text{log}(x)\), asks the question: 'To what power must a base number be raised to get x?' When no base is specified, we use the base 10, known as the common logarithm.

For example, \(\text{log}_{10}(100) = 2\), because 10 raised to the power of 2 equals 100. In our problem, we need to find \(\text{log}_{10}(0.1741)\), meaning 'To what power must 10 be raised to get 0.1741?'. This can be directly calculated using a scientific calculator.
common logarithm
A common logarithm is a logarithm with base 10. It's widely used in scientific calculations and in many other fields because many real-world phenomena grow exponentially at a rate proportional to their size.

The common logarithm is represented simply as \(\text{log}(x)\) without any subscript. For the exercise at hand, \(\text{log}(0.1741)\) represents the power to which 10 must be raised to equal 0.1741. This power is a negative number because values between 0 and 1 are sought by negative exponents. Using a scientific calculator typically helps, as log tables are no longer commonly used.
scientific calculator
A scientific calculator is an invaluable tool for working with logarithms. These calculators have a dedicated key for the common logarithm (usually labeled 'log'). Here鈥檚 how to use it for our problem:
  • First, ensure your calculator is set to base 10 logarithms. This is standard for most calculators but double-check if unsure.
  • Next, enter the number you're finding the logarithm of. For this exercise, input 0.1741.
  • Finally, press the 'log' button.

For \(\text{log}(0.1741)\), the calculator will show approximately -0.7597. This means that 10 raised to the power of -0.7597 is about 0.1741. Remember to round to four decimal places for precision.

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Most popular questions from this chapter

A student erroneously wrote \(\log _{a}(x+y)=\log _{a} x+\log _{a} y .\) When his teacher explained that this was indeed wrong, the student claimed that he had used the distributive property. WHAT WENT WRONG?

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