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We know that if $${\log_ap > m => 0 < p < a^m,\qquad if\quad 0 < a < 1}$$
I checked for ${\log_{0.5}0 = ∞}$.
Here ${a=0.5, p=0}$ and let ${m =1}$.
I get ${0 ≤ 0 < 0.5 = 0 ≤ 0 < 0.5}$.

Here upon the equality being added in ${0 ≤ 0 < 0.5}$, is the original property getting violated which only includes strict inequality?

Shreevathsa K
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2 Answers2

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The function $\log_ap$ is defined for $p > 0$ only. So writing $\log_{0.5} 0 $ does not make sense and any further inequalites do not hold.

Infinity_hunter
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According to definition of logarithm in $ log_b{x} $, $x$ must be greater than 0 or $x>0$. Thus $log _{0.5} {0}$ is wrong as here $x$ is not greater than $1$.