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!