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_yz   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 = 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 = 10^0.3010. 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 “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

aⁿ × a^m = a^{(n + m)},

so that log_a(aⁿ) = n, log_a(a^m) =m and log_a(a^{n+ m}) = n + m . Now let’s write p = aⁿ and q = a^m. Then

log_a(p × q) = n + m = log_ap + log_aq.

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^-m (post 18.2), it follows that

aⁿ/a^m = aⁿ × a^-m = a^{(n – m)}

so that

log_a(p/q) = n – m = log_ap – log_aq.

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^-m is that log_a(1/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

Follow-up posts

21.5 Logarithmic spirals