# 18.3 Logarithms 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.

Remember that

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(nm)

so that

loga(p/q) = nm = 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.

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