# Physics: Logarithms and exponentials

Syllabus: PC_BK_03

This page covers what logarithms and exonentials are, then we have a short video on negative exponentials that is a popular viva question for primary and final exams!

##### What are logarithms?

They basically answer: How many of one number do we multiply to get another number?

How many 2s do we multiply to get 8?

Answer: 2 × 2 × 2 = 8, so we had to multiply 3 of the 2s to get 8.

So the logarithm is 3.

We write “the number of 2s we need to multiply to get 8 is 3” as:

log2(8) = 3

So these two things are the same:

##### Exponents and Logarithms are related, let’s find out how …

So a logarithm answers a question like this:

In this way:

The logarithm tells us what the exponent is!

In that example the “base” is 2 and the “exponent” is 3:

So the logarithm answers the question:

What exponent do we need
(for one number to become another number)
?

The general case is:

Example: What is log10(100) … ?

102 = 100

So an exponent of 2 is needed to make 10 into 100, and:

log10(100) = 2

##### Common Logarithms: Base 10

Sometimes a logarithm is written without a base, like this:

log(100)

This usually means that the base is really 10.

It is called a “common logarithm”.

On a calculator it is the “log” button.

It is how many times we need to use 10 in a multiplication, to get our desired number.

Example: log(1000) = log10(1000) = 3

##### Natural Logarithms: Base “e”

Another base that is often used is e (Euler’s Number) which is about 2.71828.

This is called a “natural logarithm”.

On a calculator it is the “ln” button.

It is how many times we need to use “e” in a multiplication, to get our desired number.

Example: ln(7.389) = loge(7.389) ≈ 2

Because 2.718282 ≈ 7.389

##### What does this have to do with anaesthetics?

Logarithms: Best example: pH scale

Logarithmic tables like the one above shows you just how different a pH of 6.8 is from physiological pH of 7.4. At normal body pH (7.4), there are 40nm/L of hydrogen ions in the plasma. In severe illness eg. diabetic ketoacidosis, patients may have a pH of 6.8, which means that there are 158nm/L of hydrogen ions in plasma. Although there is only a difference of 0.6 in pH between someone healthy and someone critically ill, the patient with a pH of 6.8 has four times the amount of hydrogen ions in their bloodstream! A difference of 1 in pH is a 10 time difference in hydrogen ion concentrations!

Exponentials: Best example: Drug wash out elimination curves