how to find the log of 0.25 WITHOUT CALCULATOR the log of 25 is 3979 but with the zero it s is different why?

log(0.25) = log(25/100) = log(25) -log(100)
= log(25) - log(10^2)
= log(25) - 2 log(10)
= log(25)-2
=1.3979 -2
= -0.6021

Recall that there's a rule stating that the log of a product is the same as the sum of the logs:

log(xy) = log(x) + log(y)

So starting with:

log(0.25)

Turn that into the product of 25 and 10⁻²:

log(25 * 10⁻²)

Now separate them into the sum of two logs:

log(25) + log(10⁻²)

The log(25) is 1.3979 (I got from my calculator. I don' t know if what you wrote is a typo, or you just have wrong info), so make that substitution:

1.3979 + log(10⁻²)

Then we have the base 10 log of a number with a base of 10. So the 10's cancel each other out leaving only the exponent, so now we have:

1.3979 - 2

And finishing the subtraction:

-0.6021

Checking with a calculator:

log(0.25) = -0.60205999

Rounding to 4 DP, it's correct.

I hope you are learning this in a history class and not in a mathematics class. Learning how to use tables of logarithms to do arithmetic calculations was only useful before the invention of calculators, say before about 1970. I used to teach it back then!

Anyway, you have to learn about mantissas and characteristics. Mantissas are always positive, so for example

log 25 ---> mantisa = 3979, characteristic = 1 --- this means 10^1.3979 = 25
log 250 --> mantisa = 3979, characteristic = 2 --- this means 10^2.3979 = 250
log .25 ---> mantisa = 3979, characteristic = -1 -- this means 10^(-1 + .3979) = .25
log .025 -->mantisa = 3979, characteristic = -2 -- this means 10^(-2 + .3979) = .025

Comment: to avoid mixups of positive and negative it was common to add 10 to negative characteristics.

Comment2: the log of 0.25 is the same as the log of .25 because 0.25 = .25

In the following, log refers to base 10
log(2.5) ~ 0.39794
log(25) ~ 1.39794
Notice that it is 1 more. That is because
log(25) =log(10 * 2.5) = log(10^1) + log(2.5) = 1 + log(2.5)
Similarly log(250) = log(10^2) + log(2.5) = 2 + log(2.5) ~ 2.39794,
so you add one for each increasing power of 10

But you need log(0.25) which is log[10^(-1)] + log(2.5) = -1 + log(2.5)
log(0.25) = 0.39794 – 1 = ~ - 0.60206

So, it was different because numbers less than one are negative powers of 10.
In this example, 10^(- 0.60206) ~ 0.25