## anonymous one year ago Help please: log6(1/5√36)

1. anonymous

$\log_{6} \frac{ 1 }{ \sqrt[5]{36} }$

2. anonymous

So far I've done: |dw:1442966236088:dw|

3. anonymous

so... what are you asked to do with $$\bf log_{6}\left( \cfrac{ 1 }{ \sqrt[5]{36} }\right) ?$$

4. anonymous

Find x.

5. anonymous

"x"? kinda hard to find, since it isn't in the expression firstly

6. anonymous

ohh hmm just saw it ... so is $$\bf \bf log_{6}\left( \cfrac{ 1 }{ \sqrt[5]{36} }\right) =x$$ then... ok

7. anonymous

Yup, says evaluate it. My notes said to set each equation to x and solve for it.

8. anonymous

well.... can we say...turn $$\large \bf \sqrt[5]{36}$$ into a a value with a rational exponent? notice $$\large \sqrt[5]{36}\implies \sqrt[5]{6^2}$$

9. anonymous

recall that $$\large { a^{\frac{{\color{blue} n}}{{\color{red} m}}} \implies \sqrt[{\color{red} m}]{a^{\color{blue} n}} \qquad \qquad \sqrt[{\color{red} m}]{a^{\color{blue} n}}\implies a^{\frac{{\color{blue} n}}{{\color{red} m}}} \\\quad \\ % rational negative exponent a^{-\frac{{\color{blue} n}}{{\color{red} m}}} = \cfrac{1}{a^{\frac{{\color{blue} n}}{{\color{red} m}}}} \implies \cfrac{1}{\sqrt[{\color{red} m}]{a^{\color{blue} n}}}\qquad\qquad % radical denominator \cfrac{1}{\sqrt[{\color{red} m}]{a^{\color{blue} n}}}= \cfrac{1}{a^{\frac{{\color{blue} n}}{{\color{red} m}}}}\implies a^{-\frac{{\color{blue} n}}{{\color{red} m}}} }$$

10. anonymous

That makes sense. So now: 6^x = 6^2/5. Sixes cross out right? x = 2/5? Negative, since it was a fraction.

11. anonymous

close... one sec

12. anonymous

$$\bf log_{6}\left( \cfrac{ 1 }{ \sqrt[5]{36} }\right) =x\implies log_6\left( \cfrac{1}{6^{\frac{2}{5}}} \right)=x\implies log_{\color{red}{ 6}}\left( {\color{red}{ 6}}^{-\frac{2}{5}} \right)=x \\ \quad \\ \textit{log cancellation rule } \qquad log_{\color{red}{ a}}{\color{red}{ a}}^x\implies x\qquad \qquad {\color{red}{ a}}^{log_{\color{red}{ a}}x}=x\qquad thus \\ \quad \\ -\cfrac{2}{5}=x$$

13. anonymous

Thanks!