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Thread: Exponential Functions

  1. #1
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    Exponential Functions

    Please check if my answers are correct, if not, please help me by correcting them! Thanks in advance.

    Find all real numbers, x, that satisfy $\displaystyle 6^x 7^{2x} = 86,436$
    Here's what I think...
    $\displaystyle 42^{2x^{2}} = 86,436$
    $\displaystyle 2x^2 = 2,058$
    $\displaystyle 2x = +/- \sqrt {45.365}$
    $\displaystyle x = +/- 22.68$

    ----

    The total number of hamburgers sold by a national fast-food is growing exponentially. If 3 billion had been sold by 1995 and 9 billion had been sold by 2000, how many will have been sold by 2005?
    I think there will be 27 billion sold because of $\displaystyle 3^{3}$ since 9 billion in 2000 is $\displaystyle 3^{2}$ ...
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  2. #2
    Senior Member vincisonfire's Avatar
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    What you did violates many laws. Take the log on each side.
    $\displaystyle ln(6^x 7^{2x}) = ln(86,436) $
    $\displaystyle ln(6^x) + ln(7^{2x}) = ln(86,436) $
    $\displaystyle xln(6) + 2xln(7) = ln(86,436) $
    $\displaystyle x(ln(6) + 2ln(7)) = ln(86,436) $
    Solve for x. You're gonna get a nice integer answer.
    Your second answer is right.
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  3. #3
    MHF Contributor

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    Here is an easier way. Factor the number.
    $\displaystyle 86436=2^2 3^2 7^4$.
    Recall that $\displaystyle 2^2 3^2 = 6^2$.
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  4. #4
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    Hello, Macleef!

    Find all real numbers $\displaystyle x$ that satisfy: .$\displaystyle 6^x\cdot7^{2x} \:=\:86,\!436$

    We have: .$\displaystyle 6^x\cdot(7^2)^x \:=\:86,\!436 \quad\Rightarrow\quad (6\cdot7^2)^x \:=\:85,436 $

    . . $\displaystyle 294^x \:=\:86,4376 \quad\Rightarrow\quad 294^x \:=\:294^2 \quad\Rightarrow\quad \boxed{x\:=\:2}$

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