anonymous
  • anonymous
Solve 4^(2x) = 7^(x−1).
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
freckles
  • freckles
have you tried taking ln( ) of both sides
anonymous
  • anonymous
wouldn't it be log?

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freckles
  • freckles
\[\ln(4^{2x})=\ln(7^{x-1}) \\ \\ \text{ Now use power rule } \\ 2x \ln(4)=(x-1) \ln(7)\] you can use log too if you want
freckles
  • freckles
not distribute on the right hand side
freckles
  • freckles
and put your terms with x in it on one side and your terms without x on the opposing side
anonymous
  • anonymous
1.38629436112(2x) I mean, sorry :)
freckles
  • freckles
are you allowed to approximate ?
anonymous
  • anonymous
no, not until we have the final answer
freckles
  • freckles
ok so leave that one side as 2ln(4)x and distribute on the other side recall the distributive property is a(b+c)=ab+ac
freckles
  • freckles
use distributive property here and what do you get: \[\ln(7)(x-1)=?\]
anonymous
  • anonymous
1.94591014906
anonymous
  • anonymous
1.94591014906x-1.94591014906
freckles
  • freckles
I think you mean to say ln(7)x-ln(7)
anonymous
  • anonymous
oh yes! Sorry
freckles
  • freckles
\[\ln(4^{2x})=\ln(7^{x-1}) \\ \\ \text{ Now use power rule } \\ 2x \ln(4)=(x-1) \ln(7) \\ 2 \ln(4) x=\ln(7) x- \ln(7) \]
freckles
  • freckles
now to get the x people together subtract ln(7)x on both sides
freckles
  • freckles
\[ax=bx-c \\ ax-bx=-c \\ x(a-b)=-c \\ \text{ the last step is to choose to divide by } \\ \text{ what is in front of the } x \\ \\ \text{ in this example that would be } (a-b) \\ x=\frac{-c}{a-b}\]
freckles
  • freckles
this is exactly what you are going to do here first subtract ln(7)x on both sides then use my example as a guide sorta to finally find x
anonymous
  • anonymous
2.77258872224x=1.94591014906x-1.94591014906
freckles
  • freckles
I thought you aren't allowed to approximate until you have final answer
anonymous
  • anonymous
i didn't :)
freckles
  • freckles
ln(7) is irrational there is no way you can write out the whole number same for ln(4)
anonymous
  • anonymous
oh. Sorry, I just typed it into my calculator & thats what it said
freckles
  • freckles
your calculator can only show you so many digits
freckles
  • freckles
it doesn't have an infinite screen display
freckles
  • freckles
anyways... \[\ln(4^{2x})=\ln(7^{x-1}) \\ \\ \text{ Now use power rule } \\ 2x \ln(4)=(x-1) \ln(7) \\ 2 \ln(4) x=\ln(7) x- \ln(7) \\ \text{ the step I was asking you \to do is subtract } \ln(7) x \text{ on both sides } \\ 2 \ln(4) x -\ln(7) x=-\ln(7)\] try factoring the x out on the left hand side like I did in my example ax-bx=x(a-b)
anonymous
  • anonymous
2 In(x)-In(-3)x=-In (7)
freckles
  • freckles
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freckles
  • freckles
\[\ln(4^{2x})=\ln(7^{x-1}) \\ \\ \text{ Now use power rule } \\ 2x \ln(4)=(x-1) \ln(7) \\ 2 \ln(4) x=\ln(7) x- \ln(7) \\ \text{ the step I was asking you \to do is subtract } \ln(7) x \text{ on both sides } \\ 2 \ln(4) x -\ln(7) x=-\ln(7) \\ ( 2 \ln(4)-\ln(7) )x=-\ln(7)\] I'm going to leave the last step to you
anonymous
  • anonymous
Alright, so I got 2 In (-3) =-In(7) In(-3)^2=-In(7) yes?
anonymous
  • anonymous
2.35389<---is my answer
anonymous
  • anonymous
yes?
anonymous
  • anonymous
I gotta go eat lunch:) Please message me if I was wrong
freckles
  • freckles
not sure where you got 2 ln(-3)=-ln(7) just divide both sides by what x is being multiplied by which is 2ln(4)-ln(7) this does not equal 2 ln(-3) divide both sides by 2ln(4)-ln(7)
freckles
  • freckles
\[\ln(4^{2x})=\ln(7^{x-1}) \\ \\ \text{ Now use power rule } \\ 2x \ln(4)=(x-1) \ln(7) \\ 2 \ln(4) x=\ln(7) x- \ln(7) \\ \text{ the step I was asking you \to do is subtract } \ln(7) x \text{ on both sides } \\ 2 \ln(4) x -\ln(7) x=-\ln(7) \\ ( 2 \ln(4)-\ln(7) )x=-\ln(7) \\ x=\frac{-\ln(7)}{2 \ln(4)-\ln(7)}\]
anonymous
  • anonymous
-3.29?
freckles
  • freckles
http://www.wolframalpha.com/input/?i=-ln%287%29%2F%282ln%284%29-ln%287%29%29 this calculator says that is approximately -2.35

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