I have a simple question: in differential equations, it has been common in several of my homework problems to raise a base $e$ to the power of both sides of an equation to get variables out of natural log functions.
My question is what happens if you have a zero term? For example:
$
ln(x) + ln(y) = z
$
is equivalent to
$
e^{ln(x)} + e^{ln(y)} = e^z
$
which simplifies to
$
x + y = e^z
$
But what happens if $y=1$?
$
ln(x) + ln(1) = z
$
$
ln(x) + 0 = z
$
At this point, if you removed zero from the equation and then raised both sides by $e$, you would get $x = e^z$ when it should be $x + 1 = e^z$. Thinking about it in this manner, one could claim the following:
$
1 + 0 = 1
$
$
e^1 + e^0 = e^1
$
$
e + 1 = e
$
$
1 = 0
$
Is there some rule that explains why this algebra breaks down, and when it is wrong to raise both sides of an equation by $e$? This thought experiment is obviously incorrect, but in my first example, it is incorrect if you don't include zero in your raising-by-e operation.
