## anonymous one year ago Solve 4^(2x) = 7^(x−1).

1. anonymous

@Astrophysics @rishavraj @ganeshie8

2. freckles

have you tried taking ln( ) of both sides

3. anonymous

wouldn't it be log?

4. 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

5. freckles

not distribute on the right hand side

6. freckles

and put your terms with x in it on one side and your terms without x on the opposing side

7. anonymous

1.38629436112(2x) I mean, sorry :)

8. freckles

are you allowed to approximate ?

9. anonymous

no, not until we have the final answer

10. 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

11. freckles

use distributive property here and what do you get: $\ln(7)(x-1)=?$

12. anonymous

1.94591014906

13. anonymous

1.94591014906x-1.94591014906

14. freckles

I think you mean to say ln(7)x-ln(7)

15. anonymous

oh yes! Sorry

16. freckles

$\ln(4^{2x})=\ln(7^{x-1}) \\ \\ \text{ Now use power rule } \\ 2x \ln(4)=(x-1) \ln(7) \\ 2 \ln(4) x=\ln(7) x- \ln(7)$

17. freckles

now to get the x people together subtract ln(7)x on both sides

18. freckles

$ax=bx-c \\ ax-bx=-c \\ x(a-b)=-c \\ \text{ the last step is to choose to divide by } \\ \text{ what is in front of the } x \\ \\ \text{ in this example that would be } (a-b) \\ x=\frac{-c}{a-b}$

19. 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

20. anonymous

2.77258872224x=1.94591014906x-1.94591014906

21. freckles

I thought you aren't allowed to approximate until you have final answer

22. anonymous

i didn't :)

23. freckles

ln(7) is irrational there is no way you can write out the whole number same for ln(4)

24. anonymous

oh. Sorry, I just typed it into my calculator & thats what it said

25. freckles

your calculator can only show you so many digits

26. freckles

it doesn't have an infinite screen display

27. 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)

28. anonymous

2 In(x)-In(-3)x=-In (7)

29. freckles

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30. 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

31. anonymous

Alright, so I got 2 In (-3) =-In(7) In(-3)^2=-In(7) yes?

32. anonymous

33. anonymous

yes?

34. anonymous

I gotta go eat lunch:) Please message me if I was wrong

35. 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)

36. freckles

$\ln(4^{2x})=\ln(7^{x-1}) \\ \\ \text{ Now use power rule } \\ 2x \ln(4)=(x-1) \ln(7) \\ 2 \ln(4) x=\ln(7) x- \ln(7) \\ \text{ the step I was asking you \to do is subtract } \ln(7) x \text{ on both sides } \\ 2 \ln(4) x -\ln(7) x=-\ln(7) \\ ( 2 \ln(4)-\ln(7) )x=-\ln(7) \\ x=\frac{-\ln(7)}{2 \ln(4)-\ln(7)}$

37. anonymous

-3.29?

38. 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