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Math Help - logarithms/calculator/solving

  1. #1
    Newbie crazydaizy78's Avatar
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    logarithms/calculator/solving

    Can't I solve these using my calculator?
    If not, can you help me get started? I'm drawing a total blank. Thanks!!

    4^x=1/256

    and

    10^log(3x)=4
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by crazydaizy78 View Post
    Can't I solve these using my calculator?
    If not, can you help me get started? I'm drawing a total blank. Thanks!!

    4^x=1/256
    4^x = \frac{1}{256}

    It helps if you know that 256 = 4^4. Then
    4^x = \frac{1}{4^4} = 4^{-4}

    so x = -4.

    To do this if you don't know that little fact:
    4^x = \frac{1}{256}

    log_4(4^x) = log_4 \left ( \frac{1}{256} \right ) <-- If you use log_{10} or ln here then you don't have to use the change of base formula a little further on.

    x = log_4(1) - log_4(256)

    x = -log_4(256)

    Now let's say you still haven't seen that 256 = 4^4. Use the change of base formula to change the base to e (or 10 if you desire):
    log_a(b) = \frac{ln(a)}{ln(b)}

    So we have:
    x = -\frac{ln(256)}{ln(4)}
    which you can plug into your calculator and get that x = -4.

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by crazydaizy78 View Post
    10^log(3x)=4
    The ^ and log operators are inverses of each other, so in general
    a^{log_a(b)} = b = log_a(a^b)

    thus, taking "log" to be log base 10:
    10^{log(3x)} = 4

    3x = 4
    which I'm sure you can solve from here.

    -Dan
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