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Math Help - logarithm

  1. #1
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    logarithm

     \ln (r^ { 2 } s^ { 8 } \sqrt[ 8 ]{ r^ { 5 } s^ { 2 } } ) is equal to  A \ln r + B \ln s

    find A & B

    im not really sure what to do on this problem, but split up stuff inside
     \ln (r^ { 2 }) + \ln (s^ { 8 }) + \ln(\sqrt[ 8 ]{ r^ { 5 })} + \ln(\sqrt[8]{ s^ { 2 } } )

     2\ln r + 8\ln s + \frac{1}{8}(r^5) + \frac{1}{8}(s^2)

    a lil stuck at this part
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by viet View Post
     \ln (r^ { 2 } s^ { 8 } \sqrt[ 8 ]{ r^ { 5 } s^ { 2 } } ) is equal to  A \ln r + B \ln s

    find A & B

    im not really sure what to do on this problem, but split up stuff inside
     \ln (r^ { 2 }) + \ln (s^ { 8 }) + \ln(\sqrt[ 8 ]{ r^ { 5 })} + \ln(\sqrt[8]{ s^ { 2 } } )

     2\ln r + 8\ln s + \frac{1}{8}(r^5) + \frac{1}{8}(s^2)

    a lil stuck at this part
    Whoah, not so fast there. don't split everything up so quick, simplify first.

    \ln (r^2 s^8 \sqrt [8]{r^5 s^2}) = \ln (r^2 s^8 (r^5 s^2)^{ \frac {1}{8}})
    .................... = \ln (r^2 s^8 r^{ \frac {5}{8}} s^{ \frac {1}{4}})
    .................... = \ln (r^{ \frac {21}{8}} s^{ \frac {33}{4}})

    i think you can take it from here
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  3. #3
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    Quote Originally Posted by viet View Post
     \ln (r^ { 2 } s^ { 8 } \sqrt[ 8 ]{ r^ { 5 } s^ { 2 } } ) is equal to  A \ln r + B \ln s

    find A & B

    im not really sure what to do on this problem, but split up stuff inside
     \ln (r^ { 2 }) + \ln (s^ { 8 }) + \ln(\sqrt[ 8 ]{ r^ { 5 })} + \ln(\sqrt[8]{ s^ { 2 } } )

     2\ln r + 8\ln s + \frac{1}{8}(r^5) + \frac{1}{8}(s^2)

    a lil stuck at this part
    Hello, Viet,

    you nearly missed the correct solution...

    I take your last line and add the missing parts:

     2\ln r+8\ln s+\frac{1}{8}\ln {(r^5)} + \frac{1}{8}\ln {(s^2)} = 2\ln r+8\ln s+\frac{5}{8}\ln {(r)} + \frac{2}{8}\ln {(s)} =
    \left(2 + \frac{5}{8}  \right)\ln(r) + \left(8 + \frac{1}{4}  \right)\ln(s)

    I'll leave the rest for you
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