A student argues: 'Since $lo g x^{2} = 2 lo g x$ , the equation $lo g x^{2} = 4$ has the same solutions as $2 lo g x = 4$ , namely $x = 100$ (using base 10).' What is wrong with this argument?

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A student argues: *'Let $a = b$ . Then $a^{2} = ab$ , so $a^{2} - b^{2} = ab - b^{2}$ . Factorising, $(a + b) (a - b) = b (a - b)$ . Dividing by $(a - b)$ gives…
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A student argues: 'If it is raining then the ground is wet. The ground is wet. Therefore it is raining.' What is wrong with this argument?