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Math Help - Natural log rule explination pleaseee

  1. #1
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    Jul 2010
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    Natural log rule explination pleaseee

    Im really confused with the way my texbook explains the answer to this question:
    lim as x approaches 0 of (5^x- 3^x)/x
    So clearly thats impossible since there cant be a 0 in the denomintor so we use L'hospital's rule. Now the back of my textbook uses ln and this is where it got me confused:

    lim (5^x- 3^x)/x = lim (5^xln5- 3^xln3)/1
    I thought the general log rule was ln5^x = xln5 not 5^xln5 ???
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  2. #2
    Senior Member
    Joined
    Mar 2009
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    Well, we can rewrite 5^x as e^{x\cdot \ln 5}. Now
     \frac{d}{dx} 5^x = \frac{d}{dx} e^{x\cdot \ln 5} = \ln 5 \cdot e^{x\cdot \ln 5} = \ln 5 \cdot 5^x
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