Limits of Sequences with Logarithms
Krystian Karczyński
Founder and General Manager of eTrapez.
Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.
He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.
For the limits of many sequences with logarithms, you can confidently use the transformations and formulas for logarithms learned in high school. For example:
Example of a Limit of a Sequence with Logarithms
In situations where logarithms had different bases and there wasn’t much you could do about it, they were converted to a common base using the formula: 
. For our limit, it will be nice and convenient to take this base: 
. We then have the limit of the sequence:
We know that 
, which in our expression is 
. Therefore:
And this expression, using the high school formula again (but this time in reverse), will be equal to 
…
Which is of course the result (an irrational number).
There was no need to use any limit methods – just the high school logarithm transformations were enough.
Want to know more about calculating limits? I recommend my course 🙂
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