Is the unit of a value affected, when we take the log of the value?....
Example..
Log(24cm) i.e to base 10, is the cm, affected in some way?....
Although it might be log(24) , .. You know, after getting the final answer, is the unit cm affected?... Is the final value still in cm or it is unitless.??
On
The $\log$ function is defined as taking pure numbers as arguments. You need to divide your $24$ cm by something with dimensions of length before taking the log. If you just take the $\log$ of the numerical value you get $\log 1$ foot $=0, \log 30.48$ cm $\approx 1.484$. Many problems have a natural length scale, which is what you use for this.
On
$\log(24cm)$ would, by definition of logarithm, be "The number such that, when you raise $10$ to that power, you get $24cm$". How can you raise the (unitless) number $10$ to some power and get the length $24cm$ out as a result? I don't know of such a quantity. Thus I will consider $\log(24cm)$ as undefined.
The logarithm is a function between numbers, it doesn't and can not deal with units. If your equation can't be simplified to cancel the unit by $\log(\ldots \mathrm{cm} ) - \log(\ldots \mathrm{cm})$, then there is a certain mistake in your physical derivation.