If I want to find the nearest integer of $log_2(1,000,000,000)$
What I tried was to use the change of base rule for logarithms, using base 10 should simplify this down.
$$\dfrac{log_{10}(1,000,000,000)}{log_{10}(2)}$$
$$\dfrac{9}{log_{10}(2)}$$ Again the change of base, but this time to get rid of the 2 $$\dfrac{9}{(\dfrac{log_{2}(2)}{log_2(10)})}$$
$$9\cdot log_2(10)\approx 9 \cdot3 = 27$$
Although the nearest integer is 30. I'm a bit off, is there a way to improve this to actually get to 30?
$$2^{10}=1024\approx1000$$ hence $$10^9\approx(1024)^3=2^{30}.$$
As the initial relative error is $2.4\%$, after cubing the value the relative error will be close to $7.2\%$, which is not enough to be wrong by one half on the exponent.