## anonymous one year ago Solve 3^(2x) = 7^(x−1).

1. anonymous

hellpp

2. anonymous

$$\bf 3^{2x} = 7^{x-1}$$ right? for I don't see it happening

3. anonymous

yes @jdoe0001

4. anonymous

hmmm

5. anonymous

hmm hold the mayo

6. anonymous

kk

7. anonymous

@Michele_Laino

8. anonymous

hmmm I assume you've covered logarithms?

9. anonymous

First step: Take the logarithm of both sides.

10. anonymous

ao log 3^(2x)= log 7^(x-1)

11. anonymous

hello?

12. anonymous

OK. When you have a power, the way to take the log is as follows:$\log a ^{b} = b \log a$For example$\log 5^\left( x+2 \right) = \left( x+2 \right) \log 5$ Try this with your question

13. anonymous

3^(2x) log= 7^(x-1)= log

14. anonymous

$3^{2x}=7^{x-1}=7^x*7^{-1}$ $\left( 3^2 \right)^x=\frac{ 7^x }{ 7 }$ $\frac{ 9^x }{ 7^x }=\frac{ 1 }{ 7 }$ $\left( \frac{ 9 }{ 7 } \right)^x=\frac{ 1 }{ 7 }$ take log and find x

15. anonymous

−7.74293 7.74293 −1 1 These are the answer choices <3

16. anonymous

Not quite. Let's look at just the left hand side. You are trying to determine$\log 3^{2x} = ?$Compare that with the general case$\log a^b = b \log a$Substitute a = 3 and b = 2x. What do you get?

17. anonymous

2x log 3

18. anonymous

was i right

19. anonymous

Correct. Now do the same thing on the right hand side. You are trying to determine$\log 7^{x-1} = ?$Applying the same rule, what do you get?

20. anonymous

x-1=log 7

21. anonymous

yes?

22. anonymous

Not exactly. In this one, a = 7 and b = x-1. Try again

23. anonymous

7 log x-1

24. anonymous

Backwards. Try again

25. anonymous

x-1 log 7??

26. anonymous

i got -7.74293 as an answer

27. anonymous

That's it. So now you done$\log 3^{2x} = \log 7^{x-1}$$2x \log 3 = \left( x-1 \right) \log 7$OK so far?

28. anonymous

yes! I am good

29. anonymous

Great. Now expand the right hand side.

30. anonymous

0.47712125472(2x)

31. anonymous

That's the left hand side. You can multiply the 0.477... by the 2.

32. anonymous

0.95424250943 sorry

33. anonymous

Excellent. The left hand side is 0.95424250944 x. Now, on to the left hand side. First thing to do is to expand it.

34. anonymous

** right hand side. Sorry

35. anonymous

0.84509804001x-0.845098804001

36. anonymous

right??

37. anonymous

Exactly well done! So now you have$0.95424250944 x = 0.84509804001 x - 0.8450904001$Can you gather up the x's on one side and solve?

38. anonymous

-0.36797678529=-0.8450904001

39. anonymous

so would the answer be 7.74293?

40. anonymous

Problem with the left hand side. Remember, you are subtracting 0.84509804001 x from both sides. Try the left hand side again.

41. anonymous

0.10914370543x

42. anonymous

That's better. So you have$0.10914370543 x = 0.84509804001$To solve for x, divide both sides by 0.10914370543. What do you get?

43. anonymous

7.74298468868 !!!!

44. anonymous

Yayyyy!! Well done!

45. anonymous

Yayyyy! If I post another question in the open section, will u answer?

46. anonymous

thanks:)

47. anonymous

Yup. You're welcome

48. anonymous

To avoid dealing with all those decimal places, most folks will leave the logs until the end, for example$2x \log 3 = (x-1)\log 7$$(2 \log 3) x = (\log 7) x - \log 7$$(2 \log 3 - \log 7) x = -\log 7$$x = \frac{ -\log 7 }{ 2 \log 3 - \log 7 }$Then plug it into your calculator