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How do I solve this logarithmic equation for $x$ without calculator?

$\log(\sqrt{x}) = \log (x-6)$

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  • $\begingroup$ Start with the premise that: if log(a) = log(b) then a = b $\endgroup$ – Penguino Oct 23 '13 at 3:40
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$\log(a) = \log(b) \implies a = b$. Hence, we have $$\sqrt{x} = x-6$$Now let $\sqrt{x}=t > 0$. We then get that $$t = t^2-6 \implies t^2 - t -6 = 0 \implies (t-3)(t+2) = 0 \implies t= -2, +3$$But we have $t=\sqrt{x} > 0$. Hence, $t=3 \implies x = 9$.

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