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Math Help - Exponential Equations

  1. #1
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    Exponential Equations

    Solve each exponential equation below. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places.

    (1) 3^(2x) + 3^(x) - 2 = 0

    (2) 2^(x + 1) = 5^(1 - 2x)

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  2. #2
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    for the first write it as
    (3^x)^2+3^x-2=0

    then set u=3^x substitute in then solve the quadratic to find u then use the substitution and logs to find x
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  3. #3
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    Easy reply

    You said to let u = 3^(x). Of course, I can select any letter of choice to represent 3^(x), right?
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  4. #4
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    Quote Originally Posted by magentarita View Post
    You said to let u = 3^(x). Of course, I can select any letter of choice to represent 3^(x), right?
    of course you can choose any letter under the sun
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  5. #5
    A riddle wrapped in an enigma
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    Quote Originally Posted by magentarita View Post
    You said to let u = 3^(x). Of course, I can select any letter of choice to represent 3^(x), right?
    Correct, any letter will do.

    (2)

    2^{x+1}=5^{1-2x}

    \log
    2^{x+1}=\log5^{1-2x}

    (x+1)\log2=(1-2x)\log5

    x\log2+\log2=\log5-2x\log5

    x\log2+2x\log5=\log5-\log2

    x(\log2+2\log5)=\log5-\log2

    x=\frac{\log5-\log2}{2\log5+\log2}

    x\approx.234
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  6. #6
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    Hello, magentarita!

    There are a number of ways to solve #2 . . .


    Solve the following equation. Express irrational solutions
    in exact form and as a decimal rounded to 3 decimal places.

    2)\;\;2^{x + 1} \;=\; 5^{1 - 2x}

    We have: . 2^x\cdot2^1 \;=\;5^1\cdot5^{-2x} \quad\Rightarrow\quad 2^x\cdot5^{2x} \;=\;\frac{5}{2} \quad\Rightarrow\quad 2^x\cdot(5^2)^x\;=\;\frac{5}{2}

    . . =\; 2^x\cdot25^x \;=\;\frac{5}{2} \quad\Rightarrow\quad (2\cdot25)^x \;=\;\frac{5}{2} \quad\Rightarrow\quad 50^x \;=\;2.5

    Take logs: . \ln(50^x) \;=\;\ln(2.5) \quad\Rightarrow\quad x\cdot\ln(50) \;=\;\ln(2.5)


    . . Therefore: . x \;=\;\boxed{\frac{\ln(2.5)}{\ln(50)}} \;\approx\;\boxed{0.234}

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  7. #7
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    Thanks to all...Soroban

    Thank you all. Soroban....I have seen you in other math forums.
    Your name is familiar.
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