## anonymous 5 years ago Solve for x exactly (no graphing) in (ln x)^4 = ln x^4. You will get two answers. Their sum is approximately A) 4.291 B) 4.691 C) 5.091 D) 5.491 E) 5.891

1. anonymous

2. anonymous

If you just want to get the answer, use wolframalpha. If you want to learn you'll need to put in some effort.

3. anonymous

Actually I have no idea how to solve this one. Im not sure what they want. Any ideas?

4. anonymous

Personally I would start with a substitution to make it easier to see what's going on

5. anonymous

For example, lets let k = ln(x)

6. anonymous

What would we have for our new version of the equation?

7. anonymous

(k)^4= k^4

8. anonymous

not quite.

9. anonymous

Thats basically the same thing on both sides of the equal sign

10. anonymous

It should be $$k^4 = ln(x^4)$$

11. anonymous

because in general for some value a, $ln(a^4) \ne (ln\ a)^4$

12. anonymous

But there is something we can do with the x^4 because of the properties of logarithms

13. anonymous

k^4= 4ln(x) ?

14. anonymous

Yes

15. anonymous

And since we said k = ln(x) ?

16. anonymous

k^4 = 4k?

17. anonymous

indeed

18. anonymous

So solve for k

19. anonymous

do I divide by 4 first or do I take the 4th root?

20. anonymous

I would move it over to the same side so it equals 0.

21. anonymous

then factor it

22. anonymous

Actually that might not work out well

23. anonymous

I mean it's correct, but there might be something easier

24. anonymous

hmmmm im trying to factor it and im at k(k^3-4)

25. anonymous

Right, so either k = 0 or k^3 = 4

26. anonymous

That'll work ok

27. anonymous

So now lets go back to our definition of k

28. anonymous

ln(x) = 0 and ln(x)^3=4

29. anonymous

ln(x) = 0 & ln(x) = (cube root)4 ?

30. anonymous

Yep, now raise e to the power of both sides to get rid of the ln

31. anonymous

x= (cuberoot 4) e = 4.31?

32. anonymous

Should be: $x = e^0 \text{ or } x = e^\sqrt[3]{4}$

33. anonymous

Because we had $ln(x) = 0 \text{ or } ln(x) = \sqrt[3]{4}$ and $ln(a) = b \iff e^b = a$

34. anonymous

5.891 is the answer. That was a tuffy for me.

35. anonymous

Thanks

36. anonymous

Indeed. Takes a bit of thinking

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