When Napier first published his tables, he could not have foreseen the amazing breadth of applications logs would have in today's society.
From earthquake strengths to the computations that allow faxes to be transmitted, logs are being used in more ways than were initially speculated. It also has noticeable affect in many sciences including chemistry.
One use for the natural logarithm (loge) is in the calculations relating the half lives of alpha emitting sources to the energy of their particles. This is possible, as all atoms in the same alpha source emit their particles with exactly the same energy, unlike beta which releases a spread of values. By using the formula shown below, the logarithm enables the half life to be found for an unknown alpha source:
1/t = A loge +B where t = half life
A = constant
B = constant
loge = natural log
= measured in electron volts
As a result, it can be seen that particles with higher energies have a shorter half life as they are inversely proportional to each other :
half life is equal to the particle energy (A loge +B)
half life is equal to particle energy
In many situations knowing the level of noise and in particular its intensity becomes very important. In places where the level rises above 90 decibels (dB) it can be considered noise pollution, and can cause damage to hearing if exposure lasts for long periods of time. The intensity must be known as it is taken into consideration when deciding the level of protection needed. A simple way of calculating the intensity is with the formula:
Intensity = 10 log10 (I / Io) where I = intensity in Wm-2
Io = threshold of hearing
= usually taken as 1 x 10-12 Wm-2
If a sound had an intensity of 0.001 Wm-2
Intensity = 10 log10 (0.001 / 1 x 10-12)
= 10 log10 1000000000
= 90 dB or 9 bels ( 1 bel is equal to 10 decibels)
By dividing the natural logarithm (loge) by the interest rate the number of years a loan will take to double can be found out
Raising to an exponential is the opposite process of taking a log.
For example two quantities which change at a proportional rate to each other
This is usually taken as the inverse of the natural log or loge as rearranging with one of the rules leaves:
y = e*
It can be found in many processes such as cell growth, radioactive decay and compounding of interest.
The logarithmic spiral is formed from the equation r = me*, where the constant m is greater than zero, the exponential (e)
A brief history of John Napier Rules of Logarithms Examples of Logarithms
Chemical uses of Logarithms