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Math Help - Exponents

  1. #1
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    Exponents

    if 4^x-4^(x-1)=24, What is the value of (2x)^x

    I think that x=2.5 I used my calculator and not quite sure. I also am not quite sure how to do this by hand because on my test I cannot use the calculator. I can figure out by starting at zero and going up bu 1 its .75, 3, 12, 48 so so 24 is 2 times 12 so it is half between 2 and 3 thus 2.5. There has to be an easier way. Please help me
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  2. #2
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    Quote Originally Posted by IDontunderstand View Post
    if 4^x-4^(x-1)=24, What is the value of (2x)^x

    I think that x=2.5 I used my calculator and not quite sure. I also am not quite sure how to do this by hand because on my test I cannot use the calculator. I can figure out by starting at zero and going up bu 1 its .75, 3, 12, 48 so so 24 is 2 times 12 so it is half between 2 and 3 thus 2.5. There has to be an easier way. Please help me
    4^x-4^{(x-1)}=24 \quad \Rightarrow 4^x- \{ 4^x \cdot 4^{-1} \}=24
     \Rightarrow 4^x- \frac{4^x}{4}=24 \quad \Rightarrow 4^x(1- \frac{1}{4})=24 \quad \Rightarrow 4^x \cdot \frac{3}{4}=24
     \Rightarrow 4^x = \frac{24 \times 4}{3} \quad \Rightarrow (2^2)^x=32
     \Rightarrow 2^{2x}=2^5 \quad \Rightarrow 2x=5 \quad \Rightarrow\boxed {x=\frac{5}{2}}

     \therefore (2x)^x =(2 \cdot \frac{5}{2})^{\frac{5}{2}}= 5^{\frac{5}{2}}=25 \sqrt {5}
    Last edited by CaptainBlack; October 22nd 2009 at 12:35 PM.
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  3. #3
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    4^x - 4^(x-1) = 24
    4^x - 4^x(4^-1) = 24
    4^x(1 - 4^-1) = 24
    4^x(1 - 1/4) = 24
    4^x(3/4) = 24
    4^x = 24 / (3/4)
    4^x = 32
    x = log(32) / log(4) = 2.5
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