SOLUTION: finding the exact value of log1341(81) over log1341(1/3)

Question 592768: finding the exact value of log1341(81) over log1341(1/3)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
The problem gives a hint when its tells you to find an "exact value". This tells us that we cannot use a calculator (which provides decimal approximations most of the time).

So we have to look at

to see if we can either find these logs "by hand" or if we can use one or more properties of logarithms to transform this expression into one whose logs we can find "by hand".

There are probably other ways to do this problem but what I see is:

logs that are not easily found "by hand"; and
An expression that matches the pattern of the right-hand side of the change of base formula:

The change of base formula is used most often to rewrite a single log (the left-hand side) into a quotient of logs of a different base (the right-hand side). But formulas can be used in both directions. And we can use it to rewrite your given expression (using "a" = 1/3, "b" = 1341, and "p" = 81:

If you know your exponents and powers of 3 well this single log can be done "by hand":

If you don't see this, then you can use the change of base formula to change into base 3 logarithms. (We use base 3 because both 1/3 and 81 are powers of 3.):

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