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Math Help - Simplifying logarithm problem

  1. #1
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    Simplifying logarithm problem

    2log_{4}9 - log_{2}3 is what i need to simplify
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by zooply View Post
    2log_{4}9 - log_{2}3 is what i need to simplify

    Note that:

    log_{2}3 = \frac{log_{10} 3}{log_{10} 2} = 2 \frac{log_{10} 3}{2 log_{10} 2} = 2 \frac{log_{10} 3}{log_{10} 4} =2 log_{4} 3 = log_{4} 9

    Does this help?
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  3. #3
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    Quote Originally Posted by zooply View Post
    2log_{4}9 - log_{2}3 is what i need to simplify
    You need to use the change of base rule

    \log_b{x} = \frac{\log_k{x}}{\log_k{b}}.


    I always use natural logs, but you can choose whatever base you like.

    2\log_4{9} - \log_2{3} = \frac{2\ln{9}}{\ln{4}} - \frac{\ln{3}}{\ln{2}}

     = \frac{4\ln{3}}{2\ln{2}} - \frac{\ln{3}}{\ln{2}}

     = \frac{2\ln{3}}{\ln{2}} - \frac{\ln{3}}{\ln{2}}

     = \frac{\ln{3}}{\ln{2}}

     = \log_2{3}.
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