Question

what are the two values of x that satisfy log_x 2=log_3 X

Answer 1

can someone help me

Answer 2

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

Answer 3

\[2 = log_3x \iff\ ?\]

Answer 4

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..

Answer 5

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

Answer 6

Wait, where's the 2 x's

Answer 7

I only see one.

Answer 8

log_x 2=log_3 X

Answer 9

oh, that's different.

Answer 10

But even so, the rule still applies.

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

So that I can set them equal.

Answer 15

oh~

Answer 16

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

Answer 17

so k = ?

Answer 18

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

Answer 19

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

Answer 20

oh, ok. got it.

Answer 21

Also \[k = \pm \sqrt{(\frac{ln 2}{ln 3})}\]

Which is where you get your two solutions from.

Answer 22

Though you still have to solve for x.

Answer 23

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

Answer 24

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

Answer 25

\[k = log_3 x\] Remeber?

Answer 26

haha right~

Answer 27

So \[x = 3^k\]

Answer 28

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

Answer 29

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

Answer 30

It's ln2/ln3, but otherwise yes.

Answer 31

YAY <3