How to evaluate log 27 81

Question

How to evaluate log 27  81

fanduekisses · Answer

I set it to exponential form first right?

anonymous · Answer

$$\log_{81} 27$$

fanduekisses · Answer

I thought exp form was= $$27^{x}=81$$

anonymous · Answer

I was restating the problem, sorry for the confusion. You've got it right so far, do you know what to do next?

fanduekisses · Answer

Idk I'm guessing x= a fraction but idk I'm a little confused lol

fanduekisses · Answer

Oops btw I meant $$\log _{27}81$$

anonymous · Answer

Okay, that's a little different. That would be the log of 81 with base 27. You have the right exponential equation, though.

fanduekisses · Answer

Ok yeah sorry lol Thanks :)

anonymous · Answer

The next step should be to find the prime factorization of both 81 and 27. This way you'll know their common factors.

fanduekisses · Answer

9?

anonymous · Answer

That's a common factor, yes, but you need to know all the prime factors, and how many of them there are. Like for 27, the prime factorization would be 3 * 3 * 3

fanduekisses · Answer

Ohhh I see and for 81 3*3*3*3

anonymous · Answer

Exactly! And you can simplify these into 3^3 and 3^4, right? So now can you try to plug these into the exponential equation?

fanduekisses · Answer

How? like 1/3^3=1/3^4?

anonymous · Answer

Substituting 3^3 for 27 and 3^4 for 81.
$$(3^3)^x=3^4$$
Can you simplify that?

fanduekisses · Answer

the 3's cancel out? so 2x=4=2?

anonymous · Answer

I think you've got the right idea, except I'm not sure where you got the 2 from?

fanduekisses · Answer

ooooh I was thinking of it like 2x=4 divide 2 to both sides. But if that is wrong then what happens after the 3's cancel each other out?

anonymous · Answer

Oh, the first exponent is a 3, not 2.

fanduekisses · Answer

ohh

fanduekisses · Answer

so x= 4/3? :)

anonymous · Answer

Exactly, great job! :)

fanduekisses · Answer

Yay :D Thanks so much^_^ ♥

anonymous · Answer

No problem!