What power do you raise 3 to to get 9

[m0286]

Hello again..
Alright I am now on the part of 12th grade calculus, dealing with logs and exponents and stuff. I understand if say log39 That its basically what power do you raise 3 to to get 9. Well a question i need to answer is log162. So what power to I raise 16 to to get 2? I used my calculator just trying to find a number that would work, and i got 0.25. How would I show work for that? all of the solutions they gave for different equations worked different like for the log39They showed 2 ways:
3^x=9
3^x=3^2
x=2
OR
log39=log3(3^2)
=2
None of these solutions work for 16 since 2 is 2^1 and 16^1 is 16 not 2... I am confused.. Is 0.25 even the right answer.
Any Help is appreciated greatly!
Thanks ya!

 

[inha]

hint: 16=2^4

 

[dextercioby]

Yes.Use this property of the logarithms wrt a change of basis.

$$\log_{16}2=\frac{\ln 2}{\ln 16}=\frac{\ln 2}{4\ln 2}=\frac{1}{4}=0.25$$

Daniel.

 

[Corneo]

Have you heard of a change of base? You calculator can only do log base 10. Here is the formula

$$\log_b a = \frac {\log_c a}{\log_c b}$$ where c can be anything. But why not set that to 10 so your calculator can crunch it.

 

[chroot]

0.25 is the correct answer, because 16^(0.25) = 2.

The easiest way to compute logs like $\log_b x$ on a calculator is to compute

$$\log_b x = \frac{\log x}{\log b}$$

where the logs on the right can be any base at all (10, e, whatever). In other words, to find $\log_{16} 2$ on your calculator, punch up (log 2 / log 16).

- Warren

 

[m0286]

Thanks, but what does the In stand for...

 

[dextercioby]

I don't think so.Any (scientifical) calculator should have natural logarithm.It's eseential.

Daniel.

 

[shmoe]

Thanks, but what does the In stand for...

That's probably "ln", the "natural logarithm", and it means the logarithm to the base e=2.1718..., if you haven't met it in your class yet, it's probably not far off.

 

[chroot]

Dexter's post using natural logs can just as well be done with logs of any base, including 10.

- Warren