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B Strange logarithmic property

  1. Jan 26, 2017 #1
    I encountered this in http://calcchat.com/book/Calculus-10e/8/4/7/

    Capture.PNG


    How come the above expression equals the below?
    What I know it should be 4 ln(x/(4+sqrt(16-x^2))) which means the -1 becomes the power of that thing inside ln.

    Please help me. I really don't get it.
     
  2. jcsd
  3. Jan 26, 2017 #2

    andrewkirk

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    Provided ##x\in[-4,0)\cup (0,4]## we have
    $$\left|\frac{4+\sqrt{16-x^2}}x\right|=\frac{4+\sqrt{16-x^2}}{|x|}$$
    and
    $$\left|\frac{4-\sqrt{16-x^2}}x\right|=\frac{4-\sqrt{16-x^2}}{|x|}$$
    and that multiplying the two right-hand sides together gives 1. So they are reciprocals, hence their logs are additive inverses.
     
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