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

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

Mathematics
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can someone help me
Use the definition of the log like we did with the others here.
\[2 = log_3x \iff\ ?\]

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Other answers:

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..
I'm not sure there are 2 values for x that apply here though.
Wait, where's the 2 x's
I only see one.
log_x 2=log_3 X
oh, that's different.
But even so, the rule still applies.
\[Let\ k = log_3 x \implies 3^k = x\] But that means that \[k = log_x 2 \implies x^k = 2\]
So if we take the first equation and raise it to the power of k we have \[3^{kk} = x^k = 2\]
i get your previous answer. But why are you raising them to the pwr k??
So that I can set them equal.
oh~
Because now we can take the ln of both side and have \[k^2(ln 3) = (ln 2)\]
so k = ?
the sqroot of ln2/ln3. But how did you go from x^k=2 to ln2?
I didn't. I went from \[3^{k^2} = 2 \implies k^2(ln\ 3) = (ln\ 2)\]
oh, ok. got it.
Also \[k = \pm \sqrt{(\frac{ln 2}{ln 3})}\] Which is where you get your two solutions from.
Though you still have to solve for x.
wait, so there's two values for x too?
Yes, because there are 2 values for k, and k is related to x.
\[k = log_3 x\] Remeber?
haha right~
So \[x = 3^k\]
So the two values that satisfy that equation for x are?
3^sqroot ln3/ln2 and the other one is the same except it's the negative root of ln3/ln2. yes? (please say yes ^^:)
It's ln2/ln3, but otherwise yes.
YAY <3

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