TDSM 2.15

The derivative of [math]f(x) = \log_a x[/math] ([math]x \gt 0[/math]) is [math]\frac{1}{x\ln(a)} \gt 0[/math] [math]\forall x \gt 0[/math] so [math]\log_a x[/math] is an increasing function.

With [math]x \lt 1[/math] we have [math]f(x) \lt f(1) = \log_a 1 = 0[/math] (proved)

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However, when 0 < a < 1, [math]\log_a x[/math] is a decreasing function. it is not true.