Understanding the graph of First Order Integrated Rate Law

[JuliaPark2H]

How can the graph of the first order integrated rate law be interpreted? I understand that for the graph, ln[A] is graphed against time, and the negative slope is k, but the ln function graphed against time confuses me-- the concentration for the graph isn't one-to-one, as [A] doesn't decrease at a constant rate as time progresses. The ln[x] graph starts from -infinite and increases, but slows down as x continues to increase, so what does the linearity of the ln[A] vs time tell us?

[Chem_Mod]

A first order reaction follows exponential decay (for the reactant):

[A] = [A]₀*e^-kt

If we take ln of both sides we recover a linear equation if we plot ln[A] vs t. It is in fact the exact same equation, just manipulated mathematically. So if plotting ln[A] vs t is linear, we KNOW that plotting [A] vs t gives exponential decay curve, as dictated for 1st order