How do you evaluate log_6 (1/216)?

Sep 21, 2016

${\log}_{6} \left(\frac{1}{216}\right) = - 3$

Explanation:

Think of a log term asking a question.

${\log}_{10} 100$ can be read as..

"What index of 10 gives 100?"
"To what power must 10 be raised to equal 100?"

The answer is 2 because ${10}^{2} = 100$

${\log}_{10} 100 = 2$

It really is an advantage to know all the powers up to 1000.

Note that ${6}^{3} = 216$

Write the expression slightly differently as

${\log}_{6} \left(\frac{1}{216}\right) = {\log}_{6} \left(\frac{1}{6} ^ 3\right) = {\log}_{6} {6}^{-} 3$

The answer is now obvious..

"What index of 6 will give ${6}^{-} 3$"?

"To what power must 6 be raised to equal ${6}^{-} 3$"?

${\log}_{6} \left(\frac{1}{216}\right) = - 3$

Sep 21, 2016

${\log}_{6} \left(\frac{1}{216}\right) = - 3$.

Explanation:

If log_6 (1/216)=x, "then, "6^x=1/216=1/6^3=6^-3......["by, Defn."]

$\Rightarrow x = - 3$.

$\therefore {\log}_{6} \left(\frac{1}{216}\right) = - 3$.