Orders of magnitude
The order of magnitude is a way of describing and comparing very large or very small numbers using powers of ten.
For example, 4000 can be written using scientific notation as 4 \times 1000=4 \times 10^3.
The order of magnitude of this number is 3 as it is writing using 10^3.
To increase this by a factor of 10\,000 would be to increase it by four orders of magnitude or 10^4 resulting in 4 \times 10^7 .
Alternatively to decrease this by a factor of 100 would be to decrease it by two orders of magnitude or 10^2 resulting in 4 \times 10^1 or 40. That means this number is 100 times smaller than the original number.
Orders of magnitude are used to estimate the size of a number, which is useful for comparison.
If order of magnitude was being used to compare the weight of a mouse (19 grams) with that of a cat (3.6 \text{ kg}) we would write them in the same unit of measure, then estimate them by writing them in scientific notation using whole numbers. This table go through the steps for each animal.
| Steps | Cat | Mouse |
|---|---|---|
| \text{Original information} | 3.6 \text{ kg} | 19\text{ g} |
| \text{Convert to same units} | 3.6 \text{ kg} | 0.019 \text{ kg} |
| \text{Write using scientific notation} | 3.6 \times 10^0 | 1.9 \times 10^{-2} |
| \text{Orders of magnitude} | 0 | -2 |
The two animals differ by 2 orders of magnitude. That would mean the mouse is roughly 100 times smaller than the cat.
What if we were to use grams?
| Steps | Cat | Mouse |
|---|---|---|
| \text{Original information} | 3.6 \text{ kg} | 19\text{ g} |
| \text{Convert to same units} | 3600 \text{ g} | 19 \text{ g} |
| \text{Write using scientific notation} | 3.6 \times 10^{-3} | 1.9 \times 10^{-1} |
| \text{Orders of magnitude} | -3 | -1 |
The difference between the two powers is still 2.
The table below shows a comparison between the basic numbers, powers of 10, and order of magnitude.
Examples
Example 1
What is the order of magnitude of the number 6\times 10^9?
Example 2
A bakery sells 450 cupcakes per week.
What is the order of magnitude of the total number of cupcakes sold over 8 weeks.
For 1 order of magnitude increase a number is 10 times larger.
For 1 order of magnitude decrease a number is 10 times smaller.
If a number is written in scientific notation, the order of magnitude is the exponent.
Logarithms
The order of magnitude (power of base 10) of basic numbers form what is called a logarithm scale. This scale is used to write numbers as powers of 10 and is referred to as logarithms to the base 10 or \log_{10}. For example, 1000 can be written as 1 \times 1000=1 \times 10^3. The order of magnitude of this number is 3 so the logarithm of 1000 is \log_{10}(1000)=\log_ {10}(10^3)=3.
Here is the same table above with the logarithms of each number.
| \text{Basic number} | \text{Power of 10} | \text{Logarithm} |
|---|---|---|
| 1\,000\,000 | 10^6 | 6 |
| 100\,000 | 10^5 | 5 |
| 10\,000 | 10^4 | 4 |
| 1000 | 10^3 | 3 |
| 100 | 10^2 | 2 |
| 10 | 10^1 | 1 |
| 1 | 10^0 | 0 |
| 0.1 | 10^{-1} | -1 |
| 0.01 | 10^{-2} | -2 |
| 0.001 | 10^{-3} | -3 |
| 0.0001 | 10^{-4} | -4 |
| 0.00001 | 10^{-5} | -5 |
| 0.000\,001 | 10^{-6} | -6 |
Examples
Example 3
Consider the number 0.000\,001.
Express the number as a power of 10.
Find the base ten logarithm of the number.
Example 4
Use your calculator to evaluate \log _{10}80\,000.
Round your answer to four decimal places.
Example 5
Use your calculator to find the number such that the base ten logarithm of the number is 2.5.
Round your answer to two decimal places.
The base 10 logarithm is the order of magnitude of a number written as power of 10.