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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.

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    $\begingroup$ Try rewriting .25 as 2 to some power. $\endgroup$ – Christopher Toni Jul 23 '15 at 16:39
  • $\begingroup$ This may be a duplicate of the question How to figure out the log of a number without a calculator, which was answered with several practical methods. Of course remember that the $\log_{a}(x)$ can be converted to $\log_{2}(x)$ by dividing by $\log_{a}$(2). $\endgroup$ – user253804 Jul 23 '15 at 19:06
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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.

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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 $1$.

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