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Okay, this is my proof. I'm not exactly sure if this is a viable proof, but I think it works.
Hence, from x > 0, it is always increasing (gradient > 0)
y = lnx crosses the x-axis only once, so there is only one root.
Since x cannot be less than zero, as well as a monotonic increasing function for x > 0, and the fact that it crosses the x-axis once, then as x approaches 0 from the positive side, f(x) has to be approaching negative infinity.
Hence, from x > 0, it is always increasing (gradient > 0)
y = lnx crosses the x-axis only once, so there is only one root.
Since x cannot be less than zero, as well as a monotonic increasing function for x > 0, and the fact that it crosses the x-axis once, then as x approaches 0 from the positive side, f(x) has to be approaching negative infinity.