How do you solve for t in the equation $x = e^t - 3$ So what I did first is add 3 to both sides:
$x + 3 = e^t$
Now I don't know how to get $t$ alone, if anyone can explain it that would be great
 A: Have you heard of logarithms? It's been hip and hot since the $1600$s! It's the inverse function of the exponentiation function. We have
$$x = e^{t} - 3$$
$$x + 3 = e^{t}$$
$$\boxed{t = \ln(x + 3)}.$$
A: Notice:


*

*$$\ln(a)=\log_e(a)=\frac{\ln(a)}{\ln(e)}=\frac{\ln(a)}{1}$$

*$$\ln(a^b)=b\ln(a)\space\space\space\space\space\text{when}\space a,b\text{ are positive}$$

*$$\ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b)\space\space\space\space\space\text{when}\space a,b\text{ are positive}$$


$$x=e^t-3\Longleftrightarrow x+3=e^t\Longleftrightarrow\ln(x+3)=\ln\left(e^t\right)\Longleftrightarrow\ln(x+3)=t$$
And notice that $x+3\ne0$.
A: This can be done by taking logarithm of base $e$ or natural logarithm. Logarithms are basically asking a question: "To what power a number has to be raised to get another number?"
For example, since $2^3=8$ then $\log_28=3$
Now you can do the same to your equation. The solution is $t=\log_e{(x+3)}$, which is the same as $t=\ln{(x+3)}$ . Maybe you know logarithms, but if not, you would want to learn more about logarithms to know why this is true.
