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Math Help - Pre-test 6 help

  1. #1
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    Pre-test 6 help

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  2. #2
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    #6. Condense the expression to the logarithmic of a single quanity.
    [2log 5 (x+4)+7log 5 (x+7)]-1/2log 5 x


    Hello Brooke:

    Use the properties of logs.

    log(a)+log(b)=log(ab)...[1]

    log(a)-log(b)=log(a/b)...[2]

    Rewrite as:

    log_{5}(x+4)^{2}+log_{5}(x+7)^{7}-log_{5}x^{\frac{1}{2}}

    Using [1]

    log_{5}((x+4)^{2}(x+7)^{7})-log_{5}x^{\frac{1}{2}}

    Using [2]

    log_{5}(\frac{(x+4)^{2}(x+7)^{7}}{\sqrt{x}})
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  3. #3
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    Hello, Brooke!

    1. Sketch the graph of the function: f(x) \:= \:3 + e^x

    You're expected to be familiar with the graph of y\:=\:e^x.

    This is the same graph translated up 3 units.



    2. Evaluate the expression without using a calculator: \log_{128}2

    Let: \log_{128} \:=\;x

    Rewrite in exponential form: . 128^x\:=\:2

    Get the same base on both sides: . (2^7)^x\:=\:2\quad\Rightarrow\quad 2^{7x}\:=\:2^1

    Since the bases are equal, the exponents are equal: . 7x\:=\:1\quad\Rightarrow\quad x\,=\,\frac{1}{7}

    Therefore: . \log_{128}2\;=\;\frac{1}{7}



    3. Identify the logarithmic equation written in exponential form: \log_{243}81\:=\:\frac{4}{5}

    Not sure what "identify" means here . . .

    In exponential form, the expression is: . 243^{\frac{4}{5}}\:=\:81



    4. Evaluate using the change-of-base formula: \log_9517

    We have: . \log_9517\:=\:\frac{\ln517}{\ln9}\:=\:2.843606857.  ..



    5. Which is the logarithm rewritten as a ratio of natural logarithms?
    . . . A\!:\;\ln\left(\frac{3}{x}\right)\qquad B\!:\;\frac{\ln 3}{\ln x}\qquad C\!: \;\ln x - \ln 3\qquad D\!:\;\frac{\ln x}{\ln 3}

    Is there a typo? . . . Both B and D are ratios of natural logs.



    6. Condense the expression to the logarithm of a single quantity:
    . . . \left[2\log_5(x+4) + 7\log_5(x + 7)\right] - \frac{1}{2}\log_5x

    \left[2\cdot\log_5(x+4) + 7\cdot\log_5(x+7)\right] - \frac{1}{2}\cdot\log_5(x)

    . . = \;\left[\log_5(x+4)^2 + \log_5(x+7)^7\right] - \log_5\!\left(x^{\frac{1}{2}}\right)

    . . = \;\log_5\!\left[(x+4)^2\cdot(x+7)^7\right] - \log_5\!\left(x^{\frac{1}{2}}\right)

    . . = \;\log_5\!\left[\frac{(x+4)^2(x+7)^7}{x^{\frac{1}{2}}}\right]

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