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$4000 can be written using scientific notation as $4\times1000=4\times10^3$4×1000=4×103 .
- The order of magnitude of this number is $3$3 as it is writing using $10^3$103.
- To increase this by a factor of $10000$10000 would be to increase it by four orders of magnitude or $10^4$104 resulting in $4\times10^7$4×107 .
- Alternatively to decrease this by a factor of $100$100 would be to decrease it by two orders of magnitude or $10^2$102 resulting in $4\times10^1$4×101 or $40$40. That means this number is $100$100 times smaller than the original number.
For $1$1 order of magnitude increase a number is $10$10 times larger.
For $1$1 order of magnitude decrease a number is $10$10 times smaller.
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$19 grams) with that of a cat ($3.6$3.6 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 |
|---|---|---|
| Original information | $3.6$3.6 kg | $19$19 g |
| Convert to same units | $3.6$3.6 kg | $0.019$0.019 kg |
| Write using scientific notation | $3.6\times10^0$3.6×100 | $1.9\times10^{-2}$1.9×10−2 |
| Orders of magnitude | $0$0 | $-2$−2 |
They two animals differ by $2$2 orders of magnitude. That would mean the mouse is roughly $100$100 times smaller than the cat.
What if we were to use grams?
| Steps | Cat | Mouse |
|---|---|---|
| Original information | $3.6$3.6 kg | $19$19 g |
| Convert to same units | $3600$3600 kg | $19$19 kg |
| Write using scientific notation | $3.6\times10^{-3}$3.6×10−3 | $1.9\times10^{-1}$1.9×10−1 |
| Orders of magnitude | $-3$−3 | $-1$−1 |
The difference between the two powers is still $2$2.
The table below shows a comparison between the basic numbers, powers of $10$10 and order of magnitude.
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Practice questions
Question 1
What is the order of magnitude of the number $6\times10^9$6×109?
QUESTION 2
A bakery sells $450$450 cupcakes per week.
What is the order of magnitude of the total number of cupcakes sold over $8$8 weeks.
Logarithms
The order of magnitude (power of base $10$10) of basic numbers form what is called a logarithm scale. This scale is used to write numbers as powers of $10$10 and is referred to as logarithms to the base $10$10 or $\log_{10}$log10 For example, $1000$1000 can be written as $1\times1000=1\times10^3$1×1000=1×103 . The order of magnitude of this number is $3$3 so the logarithm of $1000$1000 is $\log_{10}(1000)=\log_{10}(10^3)=3$log10(1000)=log10(103)=3.
Here is the same table above with the logarithms of each number.
| Basic number | Power of $10$10 | Logarithm |
|---|---|---|
| $1000000$1000000 | $10^6$106 | $6$6 |
| $100000$100000 | $10^5$105 | $5$5 |
| $10000$10000 | $10^4$104 | $4$4 |
| $1000$1000 | $10^3$103 | $3$3 |
| $100$100 | $10^2$102 | $2$2 |
| $10$10 | $10^1$101 | $1$1 |
| $1$1 | $10^0$100 | $0$0 |
| $0.1$0.1 | $10^{-1}$10−1 | $-1$−1 |
| $0.01$0.01 | $10^{-2}$10−2 | $-2$−2 |
| $0.001$0.001 | $10^{-3}$10−3 | $-3$−3 |
| $0.0001$0.0001 | $10^{-4}$10−4 | $-4$−4 |
| $0.00001$0.00001 | $10^{-5}$10−5 | $-5$−5 |
| $0.000001$0.000001 | $10^{-6}$10−6 | $-6$−6 |
Practice questions
Question 3
Consider the number $0.000001$0.000001
Express the number as a power of $10$10.
Find the base ten logarithm of the number.
QUESTION 4
Use your calculator to evaluate $\log_{10}80000$log1080000.
Round your answer to four decimal places.
Question 5
Use your calculator to find the number such that the base ten logarithm of the number is $2.5$2.5
Round your answer to two decimal places.