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Chapter 13: Problem 7

Find the common logarithm of each of the given numbers by using a calculator. $$1.174^{-4}$$

Short Answer

Expert verified
The common logarithm of \(1.174^{-4}\) is approximately \(-0.2491\).

Step by step solution

01

Understand Common Logarithms

Common logarithms are logarithms with a base of 10. When you are asked to find the common logarithm of a number, you are essentially finding \( \log_{10} (x) \).
02

Simplify the Expression

First, simplify the expression \(1.174^{-4}\). This means the reciprocal of \(1.174\) raised to the power of 4. Use a calculator to compute this step.
03

Calculate \(1.174^{-4}\)

Using a calculator, find \(1.174^{-4}\) by dividing 1 by \(1.174^4\). This gives approximately \(0.56245\).
04

Find the Common Logarithm

Using a calculator, find \( \log_{10} (0.56245)\). This is the common logarithm of the number. The calculator should display approximately \(-0.2491\).

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Key Concepts

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

Understanding Base 10 Logarithms
Common logarithms are essential tools in mathematics and they have a base of 10, which makes them quite intuitive. The notation \( \log_{10}(x) \) is simply asking you to determine what power (or exponent) 10 must be raised to in order to yield the number \(x\) you are interested in. For instance, if you know \( \log_{10}(100) \), you'd realize it equals 2 because 10 raised to the power of 2 gives 100.
  • They simplify complex calculations, especially in fields like engineering.
  • They help in understanding the behavior of exponential growth or decay.
To wrap it up, common logarithms with a base of 10 make it easier to work through problems involving exponential changes.
Reciprocal Calculation Demystified
The reciprocal of a number is a key concept in handling expressions with negative exponents. If you have a number \(a\), its reciprocal is \( \frac{1}{a} \). When dealing with expressions like \(1.174^{-4}\), it's important to recognize that the negative exponent suggests a reciprocal.
  • This means, before applying the power, transform the number as \(1.174^{-4} = \frac{1}{1.174^4}\).
  • This method simplifies the calculation and helps achieve the correct result when using a calculator.
By understanding reciprocals, you can transform seemingly complicated expressions into manageable calculations.
Accurate Calculations with a Calculator
In today's world, calculators are essential for verifying results and ensuring accuracy. When computing mathematical expressions like \(1.174^{-4}\) and its logarithm, a calculator helps to simplify the process.
  • First, use a calculator to determine \(1.174^4\). This gives a value you can then use to find its reciprocal.
  • Next, press the division and equal buttons on your calculator to get \( \frac{1}{a} \) or the reciprocal. This will give you approximately \(0.56245\).
  • Finally, apply the logarithm function to find \( \log_{10}(0.56245) \), yielding approximately \(-0.2491\).
By following these steps, you can ensure that you're applying the calculator's functions accurately, leading to a precise final result.

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