Question

Whate is the practical application of logarithm?

Answer 1

Shown below (more calculus inlined )

Explanation:

A practical application of logarithms regarding mechanics:

You can use logarithms for models for particular real life situations

For example take a particle of mass `m` with only a resistant force of `vm` Newtons acting on it, where `v` is the velocity at time `t`

Use `F = ma => F = m (dv)/(dt)`

`=> -vm = m (dv)/(dt)`

`=> -v = (dv)/(dt)`

`=> - int 1dt = int (dv)/ v`

`=> -t + c = log_e v`

Where hence this logarithmic solution can be an appropriate model in understanding of the motion the particle has

Answer 2

Turning multiplication or division into addition or subtraction.

Explanation:

In the days before electronic calculators people used tables of logarithms to the base 10 and tables of their inverses called "anti-logarithms" to compute complicated multiplication or division using only addition or subtraction.

Using the properties of logarithms:

Let `y = a xx b`

Then `log_10 y = log_10 a + log_10 b = c` (Say)

Now, `y = 10^c` (Called the anti-logarithm of `c`)

So, with tables of `log_10 t and 10^t` one can multiply using addition.

The same is true of division using subtraction.

For historical interest see: https://www.maa.org/press/periodicals/convergence/mathematical-treasure-babbage-s-tables-of-logarithms