Finding the inverse of $y= 100(1-0.9^t)$

Question

I must solve the inverse of this function $y= 100(1-0.9^t)$ as I solve? I try but the equation $\ln$ left with negative logarithm.

Answer 1

Think about what you do to $t$ to get $y$. First you take the $t^{\text{th}}$ power of $0.9$. Then you substract the result from $1$. Finally, you multiply by $100$ and end up with $y$.

To isolate $t$ you just have to undo each of these operations in the reverse order. At no point should you be taking the logarithm of a negative.

Answer 2

$y/100 = 1- 0.9^t$, so $0.9^t = 1- y/100$, take logs of both sides using your favorite base, so $\ln(0.9)*t=\ln(1- y/100)$, etc.