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Thread: How to Solve for x in this Log Equation?

  1. #1
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    How to Solve for x in this Log Equation?

    Hello,
    I have no clue what property should be used for solving this problem or even how to solve it at all.
    Here is the equation:
    log5(log3x) = 0
    How do I solve this?
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  2. #2
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    Re: How to Solve for x in this Log Equation?

    $\log_5(\log_3(x))=0$

    $\log_3(x) = 5^0 = 1$

    $x = 3^1 = 3$

    $x=3$
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  3. #3
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    Re: How to Solve for x in this Log Equation?

    Quote Originally Posted by Locopoco View Post
    Hello,
    I have no clue what property should be used for solving this problem or even how to solve it at all.
    Here is the equation:
    log5(log3x) = 0
    How do I solve this?
    Suppose that each of $a~\&~b$ is a positive number.
    $\log_a(B)=0\iff B=1$ AND $\log_b(C)=1\iff C=b$

    Lets look at the posted question.
    $\large\log_5(\log_3(x))=0\iff\log_3(x)=1\iff x=3$
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  4. #4
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    Re: How to Solve for x in this Log Equation?

    $\log_5(\log_3(x))=0$

    Use identity,

    $\log_ab=\dfrac{\log b}{\log a}$

    $\Rightarrow\dfrac{\log\bigg(\dfrac{\log x}{\log(3)}\bigg)}{\log 5}=0$

    $\Rightarrow\log\bigg(\dfrac{\log x}{\log(3)}\bigg)=0$

    Taking exponent both sides

    $\Rightarrow \dfrac{\log x}{\log 3}=1$

    $\Rightarrow\log x=\log 3$

    $\Rightarrow x=3$
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  5. #5
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    Re: How to Solve for x in this Log Equation?

    Here is a slightly different way that involves nothing more than $log_a(b) = c \iff b = a^c$ and substitution.

    $Find\ x\ given\ log_5(log_3(x)) = 0.$

    $Let\ u = log_3(x) \implies$

    $log_5(u) = 0 \implies$

    $u = 5^0 = 1 \implies$

    $1 = u = log_3(x) \implies$

    $x = 3^1 = 3.$
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  6. #6
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    Re: How to Solve for x in this Log Equation?

    You shouldn't worry about "properties" until you have a good grasp of "definitions"! And this problem is really about the definition of "log".
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