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[cc is the most appropriate introduction and reasoning as to why teachers can find it beneficial to know and estimate some logs.]

Must we always rush for a calculator when it comes to working out pH or other calculations involving logs?

Not necessarily. Estimating values is a good skill for teachers to adopt. Most teachers and many students can see the clear relationship between logs and the power to which ten is raised ... provided they are working with 'one times' this power of ten, eg 1 x 10^–4.

What is the pH of a solution where [H⁺] = 1 x 10^–4?

Most would say pH 4 without too much effort.

Now try estimating the pH of a solution of [H+] = 2 x 10^–4? (without a calculator)

pH = –log[H⁺]. The pH = –log 2 x 10^–4 .

So we can say  pH = - (log 2 + log 10^–4 )

If you remember that log 2 = 0.3, then this and many other calculations are simple.

 pH = - (log 2 + log 10^–4 ) so substituting the log values pH = - (0.3 + (-4)) = 3.7

Are there any other logs needed to be remembered?

As well as log 2 = 0.3; it is very useful to remember that log 5 = 0.7

Remembering these two logs will enable you to have a go at estimating the value of most pH calculations or at least get a good feel for what pH is likely to be a reasonable answer.

But calculations are still required, so what is the advantage?

The calculations come down to simple addition and subtration as that is the whole benefit of using logarithms.

Remember just these two logs (log 5 = 0.7 and log 2 = 0.3) and see how you manage some simple pH calculations – without a calculator!

Try estimating the pH of solutions where the [H⁺] = 5 x 10^–5 mol dm^–3 and  [H⁺] = 2.0 mol dm^–3. You may check your estimates using your calculator if you like.