Does $\ln|x+2|=\ln|2x+4|$ and if so why so?

Question

Is $\ln|x+2|=\ln|2x+4|$? Is this right? I saw something earlier saying this was correct; my first instinct was no.

Answer 1

Its wrong. Take $x=-1$. You get $$\ln(1) = 0 \neq \ln(2).$$

Answer 2

It's almost correct. $\log |2x+4| = \log 2|x+2| = \log|x+2| + \log 2$.

Answer 3

Or just take $x=0$, then $\log 2 \ne \log 4$.

Answer 4

Is it possible what you saw before involved calculus? They may not be equal, but they both differ by a constant: $\ln|2x+4|=\ln|x+2|+\ln2$. If one were to evaluate the integral $\int\frac{2dx}{2x+4}$, either $\ln|x+2|+C$ or $\ln|2x+4|+C$ would be considered correct. But this does not mean that the $2$ logarithms are equal.