Solve for $x$ (three equations)

Question

I've been having trouble with these equations. The three equations have no relation to one and other. I've been trying to solve for $x$:

$$y = 1.15^{x+2}$$

$$y = \frac{1}{1.15^{x+2}}$$

$$y = 1 - \frac{1}{1.15^{x+2}}$$

Thank you.

Answer 1

For the first, take the $\log$ of both sides, where $\log$ means logarithm to your favourite base, perhaps $e$, perhaps $10$, perhaps $2$. We get $$\log y=(x+2)\log 1.15.$$ Thus $$x+2=\frac{\log y}{\log 1.15},$$ and now solving for $x$ is straightforward.

For the second, take the reciprocal of both sides. We get $$\frac{1}{y}=1.15^{x+2}.$$ Now use the procedure of the first problem. You may want to use the fact that $\log(1/y)=-\log y$.

Or else you can rewrite as $y=1.15^{-(x+2)}$, and take the logarithm of both sides.

For the third problem, first rewrite as $1-y=\frac{1}{1.15^{x+2}}$.