Change Of Base Theorem

Since most calculators only have base 10 and natural base e on them, logs that have different bases to these are hard to calculate.  This theorem allows us to change any log’s base to a base we can calculate easily. It states that:


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So say I want to calculate what  is.  There is no log button on the calculator that has a base of 2.  But there is one with a base of 10.  Using the above theorem:



We want to change to a base that we can use on our calculator, such as 10, so:


We can now rewrite it as follows:


The right hand side of this we can enter on our calculators, to get an answer of about 4.09.  Since 17 is just a bit more than 16, and , this answer seems to make sense.