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

- anonymous

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

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- anonymous

can someone help me

- anonymous

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

- anonymous

\[2 = log_3x \iff\ ?\]

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## More answers

- 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..

- anonymous

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

- anonymous

Wait, where's the 2 x's

- anonymous

I only see one.

- anonymous

log_x 2=log_3 X

- anonymous

oh, that's different.

- anonymous

But even so, the rule still applies.

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

So that I can set them equal.

- anonymous

oh~

- anonymous

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

- anonymous

so k = ?

- anonymous

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

- anonymous

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

- anonymous

oh, ok. got it.

- anonymous

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

- anonymous

Though you still have to solve for x.

- anonymous

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

- anonymous

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

- anonymous

\[k = log_3 x\] Remeber?

- anonymous

haha right~

- anonymous

So \[x = 3^k\]

- anonymous

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

- 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 ^^:)

- anonymous

It's ln2/ln3, but otherwise yes.

- anonymous

YAY <3

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