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Math Help - Logs

  1. #1
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    Logs

    Use Logs to expand the expression

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  2. #2
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    Quote Originally Posted by tmac11522 View Post
    Use Logs to expand the expression

    I take it by expand using logs, you mean use log laws to simplify...

    Notice

    \sqrt{z} = z^{\frac{1}{2}}

    \sqrt{y^2\sqrt{z}} = \sqrt{y^2z^{\frac{1}{2}}} = (y^2z^{\frac{1}{2}})^{\frac{1}{2}}

    \sqrt{x^4\sqrt{y^2\sqrt{z}}} = \sqrt{x^4(y^2z^{\frac{1}{2}})^{\frac{1}{2}}} = (x^4(y^2z^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{  2}}.


    So \ln{(\sqrt{x^4\sqrt{y^2\sqrt{z}}})} = \ln{((x^4(y^2z^{\frac{1}{2}})^{\frac{1}{2}})^{\fra  c{1}{2}})}

     = \frac{1}{2}\ln{x^4(y^2z^{\frac{1}{2}})^{\frac{1}{2  }}}

     = \frac{1}{2}(\ln{x^4} + \ln{(y^2z^{\frac{1}{2}})^{\frac{1}{2}}}

     = \frac{1}{2}(\ln{x^4} + \frac{1}{2}\ln{y^2z^{\frac{1}{2}}})

     = \frac{1}{2}[4\ln{x} + \frac{1}{2}(\ln{y^2} + \ln{z^{\frac{1}{2}}})]

     = \frac{1}{2}[4\ln{x} + \frac{1}{2}(2\ln{y} + \frac{1}{2}\ln{z})]

     = \frac{1}{2}(4\ln{x} + \ln{y} + \frac{1}{4}\ln{z})

     = 2\ln{x} + \frac{1}{2}\ln{y} + \frac{1}{8}\ln{z}.
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