How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$
For other problems I took the $\ln$ of the number inside the parenthesis. So for example I would've done $$y=-8\ln(1.589)^{t-3}$$ then I would get the answer $$y=-8(e^{0.46317-3})$$ but that is wrong on my assignment. What do I do with the $-3?$ I tried multiplying it by the $-8$ to get $24$ but that isn't right either.
Note: $(1.589)^{t-3}=(1.589)^t\cdot(1.589)^{-3}$. Then you have $$y=\frac{-8}{(1.589)^3}(1.589)^t$$ Now, also notice that $a=e^{\ln(a)}$. Then this gives you $$y=\frac{-8}{(1.589)^3}e^{\ln(1.589^t)}$$ Since $\ln(a^t)=t\ln(a)$, then we arrive at $$y=\frac{-8}{(1.589)^3}e^{t\ln(1.589)}$$ where $a=\ln(1.589)$ and $A_0=\frac{-8}{(1.589)^3}$