What do we mean when we talk about "powers of 10"?
You are probably familiar with the convention that 102 means (10 ×10) and 103 means (10 × 10 × 10). More generally 10n means 10 multiplied by itself n times, where n could be any number. Another way of talking about this is to refer to 10n as "10 raised to the power n".
As an example:
Here is another example. The following equation allows us to calculate a particular angle (θ):
where λ = 0.3 mm and ap = 1.5 cm. First we will convert λ to cm (using 1 mm = 0.1 cm) to make the units of length consistent:
Notice that the [cm] units on top and bottom of the equation cancel each other out and the answer therefore is a pure number with no units. This is true of functions such as the sine function. Now we will find what angle has this value for its sine by using the inverse sine (sin-1) function on a calculator. This gives us the final answer:
Logarithms
The logarithm is a function related to powers of 10. If
then
Thus the logarithm of a number (in this case the logarithm of x) is the power to which 10 must be raised to get that number. If y is an integer (whole number) this is straightforward:
The idea can also be extended to numbers other than exact powers of 10. For example
Logarithms can be calculated on a scientific calculator using the log function. (Note this is generally indicated by log or log10. It should not be confused with the "natural logarithm" function, written as either ln or loge, which is irrelevant to decibels.)
Logarithms have a number of useful properties which come directly from properties of powers of 10. Thus the logarithm of a product (i.e. two numbers multiplied together) is simply the sum of their logarithms, and a similar relationship exists for the logarithm of a ratio:
Decibels
The decibel (dB) unit of measurement is based on logarithms. It is always used to express the ratio of two quantities. For example, if the ultrasound power transmitted into the patient is P1 and the power at a certain depth is P2 then the attenuation in decibels is calculated as follows:
In diagnostic ultrasound, decibels are used as the units for attenuation, gain and dynamic range.
Table 1.4 Examples of power ratios and their value in decibels.
| P1/P2 | dB |
|---|---|
| 1 | 0 dB |
| 2 | 3 dB |
| 10 | 10 dB |
| 100 | 20 dB |
| 1000 | 30 dB |
| 0.1 | -10 dB |
| 0.01 | -20 dB |
Some of the advantages that come from using decibels are:
multiplication and division of numbers becomes addition and subtraction when the numbers are measured in decibels (since decibels are based on logarithms)
decibels make very large and very small numbers easier to manage
the calculation of attenuation becomes straightforward
the concept of dynamic range would be much more complex if decibels were not used.
Reality checking answers
It is easy to make a mistake when making calculations. It is therefore strongly recommended that you do "reality checks" whenever you can.
One way to do this is to look at your final answer and see if it seems right. You will learn, for example, that typical ultrasound wavelengths are in the range 0.1 - 1.0 mm. Thus, if you are calculating a wavelength and get an answer of 25 cm you will know that something is wrong and the calculation needs to be checked.
You can also check the accuracy of calculations themselves. As mentioned above, the use of scientific notation separates the task of the numerical calculation from that of keeping track of powers of 10. It is usually possible to do an approximate check of the numerical part of the calculation by mental arithmetic to compare with the answer you have calculated using your calculator.
In the first example in the section on scientific notation above you can approximate the calculation on the second line as (6 × 7 × 2) ÷ 3 which gives 28. You can therefore see that the answer from the calculator (23.9) is in the right ballpark.
Exercises*
1 Rearrange x = 7y to give y as a function of x.
2 Rearrange x - 3 = y + 4 to give y as a function of x.
3 Rearrange 3x + 5 = 2y - 4 to give y as a function of x.
4 Rearrange the following to give y as a function of x and z:
5 Calculate 3 + 2 - 3.
6 Calculate 3 × 2 ÷ 3.
7 Calculate:
8 Calculate the following. Simplify it first by dividing top and bottom by the same amount:
9 Calculate (a + b) given that a = 16 mm and b = 4 cm.
10 Calculate (x2 + y2) for x = 5 m and y = 20 cm.
11 Calculate (x ÷ y) for x = 1540 m/sec and y = 0.2 mm/sec.
12 Calculate 102 × 103.
13 Calculate 10-2 × 103.
14 Calculate 102 ÷ 103.
15 Write 243,000 in scientific notation.
16 Write 0.000435 in scientific notation.
17 Calculate 2820 × 0.0032 using scientific notation.
18 Calculate the following using scientific notation:
19 Use your calculator's log function to calculate log 20. Verify that this is the same as log 2 + log 10.
20 Calculate