Answer
$$y=\frac{c-e^t}{2}$$
Work Step by Step
To solve natural logarithm equations, keep in mind this property:
- If $\ln x = \ln a$ then $x=a$
$$\ln (c-2y)=t$$
- Condition: $c-2y\gt0$ or $y\lt\frac{c}{2}$
- Recall the property: $\ln e^x=x$
That means $t=\ln e^{t}$
Therefore, $$\ln(c-2y) = \ln e^{t}$$
$$c-2y=e^{t}$$
$$2y=c-e^{t}$$
$$y=\frac{c-e^t}{2}$$
We examine again the condtion that $y\lt \frac{c}{2}$:
$$y=\frac{c-e^t}{2}=\frac{c}{2}-\frac{e^t}{2}$$
We know that $e^t\gt0$, so $\frac{e^t}{2}\gt0$
That means $\frac{c}{2}-\frac{e^t}{2}\lt\frac{c}{2}$
So $y\lt\frac{c}{2}$ for all $c$ already.
