Results 1 to 5 of 5

Math Help - Solve ln(3*sqrt(x)) = sqrt (ln x)

  1. #1
    Member
    Joined
    Jun 2009
    Posts
    83

    Solve ln(3*sqrt(x)) = sqrt (ln x)

    I'm hitting a wall on this.

    It's:

    ln3 + 1/2*ln x = sqrt (ln x)

    That's where I hit the wall.
    ln 3 + 1/2*u = u^1/2

    If I square both sides it gets to be a mess.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Oct 2011
    From
    Belgium
    Posts
    36

    Re: Solve ln(3*sqrt(x)) = sqrt (ln x)

    (ln 3 + \frac{u}{2})^2 = u
    \frac{u^2}{4}+(ln3-1)u+(ln3)^2=0
    ...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2009
    Posts
    83

    Re: Solve ln(3*sqrt(x)) = sqrt (ln x)

    And then what? I don't see how to factor that, and tried the quadratic but end up with a complete mess.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2011
    From
    Belgium
    Posts
    36

    Re: Solve ln(3*sqrt(x)) = sqrt (ln x)

    Yes, it will be a mess. No alternative.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member sbhatnagar's Avatar
    Joined
    Sep 2011
    From
    New Delhi, India
    Posts
    200
    Thanks
    17

    Re: Solve ln(3*sqrt(x)) = sqrt (ln x)

    \ln{3}+\frac{1}{2}\ln{x}=\sqrt{\ln{x}}
    Let y=\ln{x} and k=\ln{3}
    \Rightarrow k+\frac{1}{2}y=\sqrt{y}

    \Rightarrow k^2+\frac{1}{4}y^2+ky=y.............[Square both the sides]

    \Rightarrow \frac{1}{4}y^2+y(k-1)+k^2=0

    \Rightarrow y=\frac{(1-k) \pm \sqrt{1-2k}}{\frac{1}{2}}.............[Quadratic Formula]

    Re substitute for y and k:
    \Rightarrow \ln{x}=\frac{(1-\ln{3}) \pm \sqrt{1-2\ln{3}}}{\frac{1}{2}}

    \Rightarrow \ln{x}=2(1-\ln{3}) \pm 2 i \sqrt{2\ln{3}-1}

    \Rightarrow x=e^{2(1-\ln{3}) + 2 i \sqrt{2\ln{3}-1}}, & e^{2(1-\ln{3}) - 2 i \sqrt{2\ln{3}-1}}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solve sqrt{x - 5} - sqrt{x - 8) = 3
    Posted in the Algebra Forum
    Replies: 4
    Last Post: July 12th 2011, 09:01 PM
  2. Replies: 2
    Last Post: September 22nd 2010, 03:48 PM
  3. Prove Q(\sqrt{2}+\sqrt{3})=Q(\sqrt{2},\sqrt{3})
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 11th 2010, 05:52 PM
  4. Replies: 11
    Last Post: January 6th 2008, 10:33 AM
  5. prove sqrt(3) + sqrt (5) is an irrational number
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: October 6th 2006, 07:48 PM

Search Tags


/mathhelpforum @mathhelpforum