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Math Help - Exponent rules with logarithms

  1. #1
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    Exponent rules with logarithms

    Hello,

    I'm having trouble remembering a basic rule.

    The question is:
    (e^x)^2 = 3

    My next step is:

    ln(e^x)^2 = ln(3)

    To my understanding, the ln and e knock each other out bringing down the x and 2. My question is, do I add them to make:

    x+2 = ln(3) ? I'm not sure about the left side of that equation.

    Thanks for any help!
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  2. #2
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    Here's what I got...so unsure about it...

    2x = ln(3)
    x=1/2 ln (3)
    x= 0.549
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  3. #3
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    Quote Originally Posted by shade View Post
    Here's what I got...so unsure about it...

    2x = ln(3)
    x=1/2 ln (3)
    x= 0.549
    Your calculations are OK. I personally prefer:

    x=\frac12 \cdot \ln(3) = \ln(\sqrt{3})
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  4. #4
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    Quote Originally Posted by shade View Post
    Hello,

    I'm having trouble remembering a basic rule.

    The question is:
    (e^x)^2 = 3

    My next step is:

    ln((e^x)^2) = ln(3)

    To my understanding, the ln and e knock each other out bringing down the x and 2. My question is, do I add them? No
    (e^x)^2 = 3~\iff~ e^{2x} = 3~\implies~2x = \ln(3)
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  5. #5
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    Hello, shade!

    Solve for x\!:\;\;(e^x)^2 \:= \:3
    Take the square root of both sides: . e^x \:=\:\sqrt{3}

    Take logs: . \ln(e^x) \:=\:\ln(\sqrt{3}) \quad\Rightarrow\quad x\cdot\ln(e) \:=\:\ln(\sqrt{3})

    Since \ln(e) = 1, we have: . x \:=\:\ln(\sqrt{3})

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  6. #6
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    Quote Originally Posted by shade View Post
    Hello,

    I'm having trouble remembering a basic rule.

    The question is:
    (e^x)^2 = 3

    My next step is:

    ln(e^x)^2 = ln(3)

    To my understanding, the ln and e knock each other out bringing down the x and 2. My question is, do I add them to make:

    x+2 = ln(3) ? I'm not sure about the left side of that equation.

    Thanks for any help!
    You are fine until this...

    ln(e^x)^2 = ln(3)

    Rules of logs say that the power ("2" in this case) gets brought down as a coefficient...

    2ln(e^x) = ln(3)

    Now, cancel out the natural log with base e

    2x = ln(3)

    Solve for x...

    x = \frac{ln(3)}{2} \Rightarrow ln(3^{\frac{1}{2}}) = ln(\sqrt{3})
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  7. #7
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    Thanks so much for everyone's help!
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