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 …
|The exponent says how many times to use the number in a multiplication.In this example: 23 = 2 × 2 × 2 = 8(2 is used 3 times in a multiplication to get 8)|
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:
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