Before you read this, I suggest you read post 18.2.
If x, y and z are numbers related by
x = yz
we say that z is the logarithm of x to the base y. This is a lot of words to explain a simple idea that is usually written
x = logyz or x = logy(z).
“Log” is a short way of writing (and saying) “logarithm”.
To make this less abstract, let’s think about logs to the base 10.
Since 100 = 102, log10(100) = 2. And, since 0.001 = 10-3, log10(0.001) = -3.
Logs need not be integers (whole numbers). In post 18.2, we saw that 2 = 100.3010. So, log10(2) = 0.3010.
Sometimes, we leave out the subscript “10” when we mean logs to the base 10. Then, log(2) means log10(2). If you look at a scientific calculator, you will usually see a button marked “log” that enables you to calculate logs to the base 10. You will probably also see a button marked “loge” or “ln” that enables you to calculate logs to the base e; e has an approximate value of 2.7183. The number e will be the subject of a later post but ex is the same as exp(x) that we met in post 16.26.
an × am = a(n + m),
so that loga(an) = n, loga(am) =m and loga(an+ m) = n + m . Now let’s write p = an and q = am. Then
loga(p × q) = n + m = logap + logaq.
So, the sum of the logs of two numbers is equal to the log of the two numbers multiplied together.
For example, the log of 6 is equal to the log of 3 added to the log of 2, for any base.
Remembering that 1/am = a-m (post 18.2), it follows that
an/am = an × a-m = a(n – m)
loga(p/q) = n – m = logap – logaq.
Then, the log of 3 is equal to the log of 6 minus the log of 2.
Another result that follows from the definition 1/am = a-m is that loga(1/m) = -loga(m).
We are now ready to return to the problem introduced in post 18.1 of how to calculate pH vales. But, before that, I want to write about another topic involving logs.
18.2 Powers of numbers
17.19 Calculating distances from speeds – integration
17.11 Measuring angles
17.4 Displacement, velocity and acceleration
16.50 Directions of forces
16.7 Writing numbers
16.6 Exponential decay
16.5 Exponential growth