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Chapter 12: Problem 28

Use a calculator to find each of the following to four decimal places. $$ \log (6)-2 $$

Short Answer

Expert verified
-1.2218

Step by step solution

01

Use Logarithm Function on Calculator

Enter the value 6 into the calculator and then press the 'log' function to find the common logarithm of 6.
02

Note the Logarithm Result

The calculator should display the result of \(\text{log}(6)\) which is approximately 0.7782.
03

Subtract 2 from the Logarithm Result

Take the result from Step 2 (0.7782) and subtract 2. Use the calculator to perform this subtraction: 0.7782 - 2.
04

Record the Final Answer

The result of the subtraction should be noted. \(\text{log}(6) - 2 \approx 0.7782 - 2 = -1.2218\).

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Key Concepts

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

Using Calculators
Calculators are essential tools for solving complex mathematical problems easily. To find the common logarithm of a number, a scientific calculator is often used. Here's a simple guide to help you use your calculator correctly for this purpose:
  • First, make sure your calculator is in the correct mode, usually denoted by a 'log' or 'LOG' button.
  • Enter the number you want to find the logarithm for. In our example, this number is 6.
  • Press the 'log' button. Your calculator will display the logarithm of the number you entered.
For example, when you enter 6 and press 'log', the display should show 0.7782. Breaking down steps like this can make using a calculator less intimidating.
Common Logarithms
Logarithms are a crucial part of algebra and higher mathematics. The logarithm used in our example is a common logarithm. Common logarithms have a base of 10 and are denoted as \(\text{log}_{10}x\). We usually just write them as \(\text{log}x\). Here are essential points to know:
  • A common logarithm tells us how many times we need to multiply 10 to get a certain number.
  • The common logarithm of 10 is 1 because 10 raised to the power of 1 is 10.
  • Similarly, \(\text{log}(100)\) is 2 because 10 raised to the power of 2 is 100.
When calculating the common logarithm of 6 using a calculator, we find it approximately equals 0.7782. This means that 10 raised to the power of 0.7782 is approximately 6.
Subtraction in Algebra
Subtraction is a fundamental arithmetic operation and is essential in algebra. Subtracting numbers helps us find the difference or balancing equations. In the context of solving logarithmic problems, subtraction can simplify an equation or expression.

In our exercise, we need to subtract 2 from the common logarithm of 6. Here's a step-by-step guide:
  • First, find the logarithm of the number using your calculator. For 6, it's approximately 0.7782.
  • Next, subtract the specified number, which is 2 in this case.
  • Using your calculator, enter 0.7782 and then perform the subtraction: 0.7782 - 2.
The result of this subtraction is -1.2218. This operation shows that even with logarithmic functions, basic arithmetic skills like subtraction play a critical role.

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Most popular questions from this chapter

Find the distance between each pair of points. $$(1,5)$$ and $$(4,1)$$

Solve by completing the square. $$ x^{2}+8 x=1 $$

Simplify. $$ \log _{10}\left(\log _{4}\left(\log _{3} 81\right)\right) $$

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b}(3 b)$$

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a} \frac{a}{\sqrt{x}}-\log _{a} \sqrt{a x}$$
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