anonymous 5 years ago what are the two values of x that satisfy log_x 2=log_3 X

1. anonymous

can someone help me

2. anonymous

Use the definition of the log like we did with the others here.

3. anonymous

$2 = log_3x \iff\ ?$

4. anonymous

x=3^(log_x2) idk....there are two xs and it's seems ambiguous....like, if i do 2=x^(log_3 X) then, it's just weird..

5. anonymous

I'm not sure there are 2 values for x that apply here though.

6. anonymous

Wait, where's the 2 x's

7. anonymous

I only see one.

8. anonymous

log_x 2=log_3 X

9. anonymous

oh, that's different.

10. anonymous

But even so, the rule still applies.

11. anonymous

$Let\ k = log_3 x \implies 3^k = x$ But that means that $k = log_x 2 \implies x^k = 2$

12. anonymous

So if we take the first equation and raise it to the power of k we have $3^{kk} = x^k = 2$

13. anonymous

i get your previous answer. But why are you raising them to the pwr k??

14. anonymous

So that I can set them equal.

15. anonymous

oh~

16. anonymous

Because now we can take the ln of both side and have $k^2(ln 3) = (ln 2)$

17. anonymous

so k = ?

18. anonymous

the sqroot of ln2/ln3. But how did you go from x^k=2 to ln2?

19. anonymous

I didn't. I went from $3^{k^2} = 2 \implies k^2(ln\ 3) = (ln\ 2)$

20. anonymous

oh, ok. got it.

21. anonymous

Also $k = \pm \sqrt{(\frac{ln 2}{ln 3})}$ Which is where you get your two solutions from.

22. anonymous

Though you still have to solve for x.

23. anonymous

wait, so there's two values for x too?

24. anonymous

Yes, because there are 2 values for k, and k is related to x.

25. anonymous

$k = log_3 x$ Remeber?

26. anonymous

haha right~

27. anonymous

So $x = 3^k$

28. anonymous

So the two values that satisfy that equation for x are?

29. anonymous

3^sqroot ln3/ln2 and the other one is the same except it's the negative root of ln3/ln2. yes? (please say yes ^^:)

30. anonymous

It's ln2/ln3, but otherwise yes.

31. anonymous

YAY <3