*Before you read this, I suggest you read* post 18.2.

If *x*, *y* and *z* are numbers related by

*x* = *y ^{z}*

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* = log* _{y}z* or

*x*= log

*(*

_{y}*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 = 10^{2}, log_{10}(100) = 2. And, since 0.001 = 10^{-3}, log_{10}(0.001) = -3.

Logs need not be integers (whole numbers). In post 18.2, we saw that 2 = 10^{0.3010}. So, log_{10}(2) = 0.3010.

Sometimes, we leave out the subscript “10” when we mean logs to the base 10. Then, log(2) means log_{10}(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 “log_{e}” 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 *e ^{x}* is the same as exp(

*x*) that we met in post 16.26.

Remember that

*a ^{n}* ×

*a*=

^{m}*a*

^{(n + m)},

so that log_{a}(*a ^{n}*) =

*n*, log

_{a}(

*a*) =

^{m}*m*and log

_{a}(

*a*) =

^{n+ m}*n*+

*m*. Now let’s write

*p*=

*a*and

^{n}*q*=

*a*. Then

^{m}log_{a}(*p* × *q*) = *n* + *m* = log_{a}*p* + log_{a}*q*.

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/*a ^{m}* =

*a*(post 18.2), it follows that

^{-m}*a ^{n}*/

*a*=

^{m}*a*×

^{n}*a*=

^{-m}*a*

^{(n – m)}

so that

log_{a}(*p*/*q*) = *n* – *m* = log_{a}*p* – log_{a}*q*.

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/*a ^{m}* =

*a*is that log

^{-m}*(1/*

_{a}*m*) = -log

*(*

_{a}*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.

__Related posts__

18.2 Powers of numbers

17.19 Calculating distances from speeds – integration

17.11 Measuring angles

17.4 Displacement, velocity and acceleration

17.2 Vectors

16.50 Directions of forces

16.7 Writing numbers

16.6 Exponential decay

16.5 Exponential growth