Logarithm on absolute values

Question

I need to solve this equation for y (C is just a constant):

$\ln(|y|) = \ln(|x|) + C$

Using the exponential function I'd get:

$|y|=|x| + e^C $

But my textbook says the solution should be:

$|y|=|x|\cdot e^C $ and thus $y =\pm xe^C$

Why is it $\cdot$ instead of +?

Answer 1

In general, $e^{a+b}=e^a\cdot e^b$. So when you apply the exponential function to $|x|+C$ you get $e^{|x|+C}=e^{|x|}\cdot e^C$.

Answer 2

Let $D$ represent the right side of your equation and $B$ represent the left side.

Then $B=D$ and $e^B=e^D$.

We know $D=ln|x|+C$ so $e^{ln|x|+C}=e^{ln|x|}e^C=|x|*e^C$

We used the fact that $e^{x+y}=e^x *e^y$

Answer 3

Why is it $*$ instead of $+$?

Because $M^{u+ w} = M^u* M^w$ and it is not true that $M^{u+w} = M^u + M^w$

Althought we are long past the point where we can thin of $M^u$ as $\underbrace{M\times M\times ....M}_{u\text{ times}}$, so $M^{u+v}=\underbrace{M\times M\times ....M}_{u+v\text{ times}} = \underbrace{M\times M\times ....M}_{u\text{ times}}\times \underbrace{M\times M\times ....M}_{v\text{ times}}$, we still have $M^{u+v}= M^u*M^v$ as a basic exponent identity.