I was wondering if there are any methods for refining the log value of numbers. Lets say I want to know the log value of 122.
I know that log(100) is 2 and I know that log(1.22) should be lower than 0.09691001 (log(6)-log(5)) and higher than 0.07918125 (log(6)-log(5)) since 1.22 is between 1.2 and 1.25.
Is there any way to get a closer estimation besides guessing that 1.22 should be near 0.086/0.087?
$$\log 122 = \log (100 * 1.22) = 2 + \log( 1.22) > 2 + \log(6 / 5) = 2 + \log(6) - \log(5) \approx 2 + 0.77815125 - 0.698970004 = 2.07918125$$
It's an okay estimate.
You could also try $1.22 < \frac{11}{9}$, if you have those two numbers; this will give you
$$\log(122) < 2 + \log(11) - \log(9) = 2 + 1.04139269 - 0.954242509 = 2.08715018$$
Actual value is $\log(122) = 2.08635983$.