Exponents, Powers of Ten and Logs

You may have come here just to find out about unit prefixes – Mega, Giga, milli etc. However, it is a good idea to understand how to multiply and divide exponents as well so the first bit is worth reading.

Notation

In physics we often come across quantities that are multiplied by itself. For example r x r . This is often written as r2.
Similarly

Properties

Multiplication


more generally

i.e. when we multiply we add the exponents.
We can only do this if the variable is the same. i.e.

Division


more generally

i.e. when we divide we subtract the exponents
This gives us an understanding of what a negative exponent means:

Therefore

It is interesting to see what happens if n = m

Any number raised to the 0 power is 1

Fractional Exponents

What is the meaning of
?
Well if we multiply two of these numbers together

So we can see that
Similarly
is the cube root of x

Powers of Ten

Interestingly
100 = 10×10 = 102
1,000,000 = 10x10x10x10x10x10 = 10 6
i.e. the exponent is the number of zeros after the one.
We can use the powers of ten to express very large or very small numbers.

We can therefore express large numbers as multiples of powers of ten. For example the speed of light is 3×108 m/s.

This makes remembering numbers quite easy. All you have to do is to remember the number and the exponent and not loads of zeros.

Also this can make calculations easier. For example
300,000,000 x 300,000,000 x 5,940,000,000,000,000,000,000,000
= 3×108 x 3×108 x 5.97 x 1024
= 3 x 3 x 5.97 x 108 x 108 x 1024
= 53.97 x 108+8+24
= 53.97 x 1040
= 5.397 x 1041

Expressing Powers of Ten on a Computer

When using computer programmes such as Excel powers of ten are expressed by entering the number then an ‘E’ and then the power of ten. For example
1,000,000 (1 x 106)can be entered as 1E6
1,234,000,000 (1.234 x 109) can be entered as 1.234E9

Unit Prefixes

All units can have a prefix to denote the magnitude of the unit compared to the base unit. We use this all the time even if we do not know it. For example the prefix for 1000 is kilo so 1 Km is 1000m and 1Kg is 1000g.

Exa 1018 1,000,000,000,000,000,000
Peta 1015 1,000,000,000,000,000
Tera 1012 1,000,000,000,000
Giga 109 1,000,000,000
Mega 106 1,000,000
Kilo 103 1,000
deci 10-1 0.1
centi 10-2 0.01
milli 10-3 0.001
micro 10-6 0.000001
nana 10-9 0.000000001

According to the standards you should only use prefixes that are powers of three – which is why I have shaded deci and centi. It does mean that centimetres are not ‘standard’ units.

Computer Units

You will get some ‘clever’ people arguing that kilo means 1024 not 1000. This comes from computing (1024 is 210). In fact the correct unit for computer memory is not kilobytes but kibibytes (although nobody uses them).

Variable in Exponent

Very often the variable is in the exponent so we have expressions such as ex. We can simply use the same rules as above for multiplying and dividing. For example
e2xe3x=e5x
At the moment just take e to be a number – however, it is really a very special number.
Let us have a look at the graph of ex and e-x

These are exponential increase and exponential decay which occur in many things such as radioactive decay, population growth and compound interest.
Compound Interest:

Radioactive Decay:

Logs – the Inverse of Exponents

If we have an equation y = bx then how do we find the value of x?
The answer is something called a logarithm.

There are different types of logs based on different bases. Base 2 (log2 or lb)is used in computing, base e (loge, log or ln) in economics, physics, mathematics and base 10 (log10 or log) in schools and some logarithm based units (see below). We nearly always arrange things so that it is expressed in one of these bases which makes things much easier.
y = ex
ln(y) = x ln(e)

However since ln(e) = 1 we have ln(y) = x

Doing Calculations Using Logs

Logarithms used to be used a lot more before calculators and computers were widely used since it is easier to add and subtract than to multiply and divide. For example if you wanted to multiply 2456.44 by 38782.23 you would look up the two numbers in log tables to get log( 2456.44) = 3.3903 and log(38782.23) = 4.5886. Add the two logs together to get 7.9789. Then you would look up the inverse log to get 95266221.06
Calculators have made things a lot easer.

Log Scales and Units

Logarithmic scales are often use to show graphs where the results cover a very wide range of values. However, the axis are often not labelled as logarithms. This can be misleading (sometimes intentionally so). You can notice a logarithmic scale because instead of the axis increasing arithmetically i.e. 1,2,3,4 or 5, 10, 15, 20 they will increase 1, 10, 100, 1000.
For example

You can also buy logarithmic paper for plotting such graphs

Log Units

There are also logarithmic units such as the decibel (which is actually one tenth of a bel – named after Alexander Graham Bell rather than a tenth of a standard ding-dong bell).
Another example of a logarithmic unit is the Richter scale for earthquakes which is the logarithm of the amplitude of the shock as recorded on the seismometer.
The pH acid/alkali scale is also logarithmic. It is the concentration of H+ ions. Pure water has a molar concentration of 10-7 and has a pH of 7. An acid could have a concentration of 10-2 – pH 2.

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