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Math Help - logarithms

  1. #1
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    logarithms

    find x if 3^(log base (a) x)+3*x^(log base(a) 3)=2

    please help
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  2. #2
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    oh i got the answer we know by property that the two logs in question would be equal

    then we can solve it
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  3. #3
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    Hello, prasum!

    I don't understand your reasoning.
    Will you tell us your answer?

    I think I solved it . . . and got an ugly answer.


    \text{Find }x\text{ if: }\;3^{\log_ax} +3\cdot x^{\log_a3} \:=\:2

    \text{Let: }\:3^{\log_ax} \:=\;P

    \text{Take logs, base 3: }\;\log_3\!\left(3^{\log_ax}\right) \;=\;\log_3(P)

    . . . . . . . . . . . \log_a(x)\cdot\underbrace{\log_3(3)}_{\text{This is 1}} \;=\;\log_3(P)
    . . . . . . . . . . . . . . . . . \log_a(x) \;=\;\dfrac{\log_a(P)}{\log_a(3)}

    . . . . . . . . . . . . . . . . . \log_a(P) \;=\;\log_a(x)\cdot\log_a(3)

    . . . . . . . . . . . . . . . . . . . . . P \;=\;a^{\log_a(x)\cdot\log_a(3)}

    . . . . . . . . . . . . . . . . . . . . . P \;=\;\left(a^{\log_a(x)}\right)^{\log_a(3)}

    . . . . . . . . . . . . . . . . . . . . . P \;=\;x^{\log_a(3)}

    Hence: . 3^{\log_a(x)} \;=\;x^{\log_a(3)}


    The equation becomes: . x^{\log_a(3)} + 3\cdot x^{\log_a(3)} \;=\;2

    . . . . . . . . . . . . . . . . . . . . . . . . 4\cdot x^{\log_a(3)} \;=\;2

    . . . . . . . . . . . . . . . . . . . . . . . . . . x^{\log_a(3)} \;=\;\dfrac{1}{2}

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x \;=\;\left(\dfrac{1}{2}\right)^{\frac{1}{\log_a(3)  }}

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x \;=\;\left(\dfrac{1}{2}\right)^{\log_3(a)}

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