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 Ã— 2 Ã— 2 = 8**, so we had to multiply 3 of the **2**s to get **8**.

So the logarithm is 3.

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

log_{2}(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: 2^{3} = 2 Ã— 2 Ã— 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 log_{10}(100) … ?

10^{2} = 100

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

log_{10}(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) = log _{10}(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) = log _{e}(7.389) â‰ˆ 2**

Because **2.71828 ^{2} â‰ˆ 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**