Exponential Function Equation and inverse Pre-Cal

Question

I am not completely sure if I wrote the equation correctly.

For A I wrote: $m(t)=100(b^x)$ Not sure it is correct...but how do I find the inverse? That doesn't make sense to me. Do I use log?

Answer 1

Notice that:

$t = 0$: $M(0) = 100 = 100(0.84^0)$

$t = 1$: $M(1) = 84 = 100(0.84^1)$

$t = 2$: $M(2) = 70.56 = 100(0.84^2)$

So that an equation modeling this relationship is $M(t) = 100(0.84^t)$ (similar to what you have written).

To find the inverse function, we can

1) replace $M(t)$ with $y$

2) solve for $t$

3) replace $y$ with $t$ and replace $t$ with $(M^{-1})(t)$:

\begin{align*} M(t) & = 100(0.84^t)\\ y & = 100(0.84^t)\\ \frac{y}{100} & = 0.84^t\\ \ln\left(\frac{y}{100}\right) & = \ln(0.84^t)\\ \ln\left(\frac{y}{100}\right) & = t \cdot \ln(0.84)\\ \frac{\ln\left(\frac{y}{100}\right)}{\ln(0.84)} & = t \end{align*}

So, \begin{align*} (M^{-1})(t) & = \frac{\ln\left(\frac{t}{100}\right)}{\ln(0.84)}\\ & = \frac{\ln y - \ln 0.84}{\ln 0.84}\\ & = \frac{\ln y}{\ln 0.84} - 1 \end{align*}