Example Evaluate following logarithms.
log_{4} 16
Solution
Now, the reality is that directly evaluating logarithms can be a very complicated process, even for those who actually understand them. Usually it is much convenient to first convert the logarithm form into exponential form. In that form usually we can get the answer pretty quickly.
What we are actually asking here is the following.
log4 16 = ?
As recommended above, let's convert this to exponential form.
log_{4} 16 = ? ⇒ 4 ^{?} = 16
Most of the people cannot evaluate the logarithm log_{4 }16 right off the top of their head. Though, most of the people can determine the exponent which we need on 4 to obtain 16 once we do the exponentiation. So, since,
4^{2} = 16
we need to have the following value of the logarithm.
log_{4} 16 = 2
Expectantly, now you have an idea on how to evaluate logarithms & are beginning to get a grasp on the notation. There are a few more evaluations which we desire to do though, we have to introduce some special logarithms which occur on extremely regular basis. They are the common logarithm & the natural logarithm. Following are the definitions & notations which we will be by using for these two logarithms.
common logarithm : log x = log_{10} x
natural logarithm : ln x =log_{e} x
Thus, the common logarithm is just the log base 10, except we drop the "base 10" part of the notation. Alike, the natural logarithm is just the log base e along with a different notation and where e is the similar number which we illustrates in the earlier section and is defined to be
e = 2.718281827.