## SugarRainbow 3 years ago how to do log base 16 of 1/4 ?!?!?!?!

1. ivanmlerner

$\log_{16}\frac{1}{4}=x \rightarrow 16^x=4^{-1}$Put them in the same base:$4^{2x}=4^{-1}$ $2x=-1\rightarrow x=-\frac{1}{2}$

2. cruffo

or use change of base formula... $\log_{16} \left(\frac{1}{4}\right) = \frac{\log 0.25}{\log 16}$

3. SugarRainbow

then what would i do how do you divide them?

4. SugarRainbow

or if i use what ivanmlerner did what would i do next?

5. cruffo

For the change of base formula, just use a calculator with a log key: log(0.25)/log(16) =

6. SugarRainbow

but i can't use a calculator

7. SugarRainbow

what if i did it how ivanmlerner did it?

8. cruffo

if you do it the way @ivanmlerner suggested, then you are done. By finding x, you found the value of the log, that is $\log_{16} \left(\frac{1}{4}\right) = -\frac{1}{2}$

9. SugarRainbow

wait so x is -1/2

10. SugarRainbow

how did you get -1/2

11. SugarRainbow

wait nevermind i got it

12. SugarRainbow

wait so what about something like log base 1/4 of 16?

13. cruffo

you would go about it the same way, basically. Asking what $$\log_{1/4} 16$$ is equal to is the same as asking the question "what power do I raise $$\dfrac{1}{4}$$ to to get $$16$$?" In math, that means solving the equation for x: $\large \left(\frac{1}{4}\right)^x = 16$

14. SugarRainbow

so 64?