anonymous
  • anonymous
what are the two values of x that satisfy log_x 2=log_3 X
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
can someone help me
anonymous
  • anonymous
Use the definition of the log like we did with the others here.
anonymous
  • anonymous
\[2 = log_3x \iff\ ?\]

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

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