how to find log base 2 of decimal number without calculator

Question

As with calculator things are simple but I don't know how to calculate log base 2 of decimal number without calculator. like $\log_2(0.25)$ etc.

Answer 1

Logarithms are easier to calculate if you can write your input as a power of the base. In this case, $\log_2(0.25) = \log_2(\frac{1}{4}) = \log_2(2^{-2}) = -2$.

In general, $\log_a(a^k) = k$. So writing the input as a power of your base gives you the easiest way to evaluate a logarithm. If the input and base aren't related by a nice power relationship, you may have to relate them to known values or use a calculator.

Answer 2

Regarding $\log_a(x)$ if $x \in (0, 1) $ you can just multiply $x$ by $a$. The amount of times you can do so without going over $1$ is the $\lfloor \log_a(x) \rfloor$.

Same with $x > 1$ except you divide by $a$ instead without going below $1$.