Solve LOG Problem: Evaluate Log 3 (Base 27) Without Calculator

[TheNovice]

1. Evaluate WITHOUT using a calculator.



2. log base 27, 3, that is, log 3 with a base of 27 instead of 10



3. I'm stuck on how you end up with the solution, which is 1/3.

Can someone provide a step by step walk through on how to solve this problem without a calculator?

 

[Kevin_Axion]

Well, any number, x, to the power of a fraction is defined such that x^\frac{a}{b} is equal to \sqrt<b>{x^a}</b>. Therefore log_{27} 3 is defined such that 27^x is equal to 3 and the only solution is 27^\frac{1}{3} and this is the same as the \sqrt[3]{27} which is equal to 3.

 

[TheNovice]

Well, any number, x, to the power of a fraction is defined such that x^\frac{a}{b} is equal to \sqrt<b>{x^a}</b>. Therefore log_{27} 3 is defined such that 27^x is equal to 3 and the only solution is 27^\frac{1}{3} and this is the same as the \sqrt[3]{27} which is equal to 3.

Thanks! that helped a lot!