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When you rearrange the equation ln [A]t = -k t + ln [A]o, you get ln [A]t-ln [A]o= -k t. Following the log rules, ln[A]t-ln[A]o is equal to ln([A]t/[A]o). Flipping the fraction to get ln([A]o/[A]t) is simply the negative, which explains the change in sign when you flip the fraction.
It has to do with the log rules, if it is organized in a way that the negatives cancel when subtracting the ln[A]t over, then you get t=(1/k) ln[A]o/ln [A]t. If you change it, you can rearrange the equation so t=-(1/k) ln[A]t/ln[A]o, and the ln[A]t is divided by the ln[A]o.
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