/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Use a calculator to evaluate the... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 37

Use a calculator to evaluate the expression, correct to four decimal places. (a) \(\log 2\) (b) \(\log 35.2\) (c) \(\log \left(\frac{2}{3}\right)\)

Short Answer

Expert verified
(a) 0.3010 (b) 1.5463 (c) -0.1761

Step by step solution

01

Understanding Logarithms

The logarithm of a number is the exponent to which the base (typically 10 for common logarithms) must be raised to produce that number. We will use a calculator with a log function to evaluate these expressions.
02

Calculate \( \log 2 \)

Use a calculator to find \( \log 2 \). Enter 2 into your calculator and press the \( \log \) button. The calculator should display the result: 0.3010 (rounded to four decimal places).
03

Calculate \( \log 35.2 \)

Enter 35.2 into your calculator and press the \( \log \) button. The result will be 1.5463, rounded to four decimal places.
04

Calculate \( \log \left(\frac{2}{3}\right) \)

To find \( \log \left(\frac{2}{3}\right) \), first calculate the fraction \( \frac{2}{3} = 0.6667 \). Then enter 0.6667 into the calculator and press the \( \log \) button. The result should be -0.1761, rounded to four decimal places.

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Key Concepts

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

Common Logarithms
Logarithms might sound complicated, but they鈥檙e actually quite straightforward when broken down. A common logarithm is a type of logarithm that uses 10 as its base. This is denoted by 'log', as opposed to \(' a \ ext {log} b \ ext {\} \)' which might use a different base. Essentially, when you see \(\log(x)\), it鈥檚 asking, 鈥淭o what power does 10 need to be raised to give x?鈥�

For instance:
  • \( \log(100) \) = 2 because \( 10^2 = 100 \).
  • \( \log(1000) \) = 3 because \( 10^3 = 1000 \).
Using this understanding, you can see that common logarithms are particularly useful in problems involving powers of ten, making calculations easier and more intuitive. They show up frequently in sciences and engineering fields.
Calculator Usage
Using a calculator to find logarithms is easy once you know the steps. Modern calculators have a specific button for common logarithms, typically labeled 'log'. This feature simplifies the process of finding logarithms without needing to manually do the math.

When using a calculator to evaluate a logarithm:
  • Enter the number you want to find the logarithm of.
  • Press the 'log' button.
  • The calculator displays the result automatically.
  • For precision, ensure that the calculator is set to give enough decimal points as required.
Let鈥檚 consider the example of calculating \( \log 35.2 \): Just enter 鈥�35.2鈥� and hit the 'log' key. The calculator will show the result, which you can round as needed. This method significantly saves time and reduces error that might occur in manual calculations.
Decimal Approximation
Calculating logarithms often results in non-whole numbers, which are usually expressed as decimal approximations. It鈥檚 essential to understand how to handle these approximations for clarity and accuracy.

When asked to approximate to a certain number of decimal places:
  • Find the full decimal value from your calculator.
  • Count the number of places after the decimal point.
  • Round the number to the required precision (e.g., four places).
For example, if your calculator gives you a value of \( 1.546259 \) for \( \log 35.2 \), rounding it to four decimal places would give \( 1.5463 \). The process ensures precision in scientific calculations and helps communicate findings effectively.

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Most popular questions from this chapter

Use a graphing device to find all solutions of the equation, rounded to two decimal places. $$e^{x^{2}}-2=x^{3}-x$$

A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time \(t\) (in hours) required for the colony to grow to \(N\) bacteria is given by $$t=3 \frac{\log (N / 50)}{\log 2}$$ Find the time required for the colony to grow to a million bacteria.

Draw the graph of \(y=3^{x},\) then use it to draw the graph of \(y=\log _{3} x\).

Simplify: \(\left(\log _{2} 5\right)\left(\log _{5} 7\right)\)

The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If \(D_{0}\) is the original amount of carbon-14 and \(D\) is the amount remaining, then the artifact's age \(A\) (in years) is given by $$A=-8267 \ln \left(\frac{D}{D_{0}}\right)$$ Find the age of an object if the amount \(D\) of carbon- 14 that remains in the object is \(73 \%\) of the original amount \(D_{0}\).
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