Why is $\ln(\sqrt{|2x-5|}) + \frac{1}{2} \ln(|2x+3|) \neq \ln(\sqrt{|2x-5|}) + \ln(\sqrt{|2x+3|})$ in Wolfram Alpha?

Question

According to Wolfram Alpha, $$\frac{1}{2} \ln(|2x+3|) = \ln(\sqrt{|2x+3|})$$

is always true, which makes sense given what I know of log rules.

However, if I add the expression $\ln(\sqrt{|2x-5|})$ to both sides of that equation, as such: $$\ln(\sqrt{|2x-5|}) + \frac{1}{2} \ln(|2x+3|) = \ln(\sqrt{|2x-5|}) + \ln(\sqrt{|2x+3|})$$

WA tells me that the two sides of this equation are not always equal! How is this possible if $\frac{1}{2} \ln(|2x+3|) = \ln(\sqrt{|2x+3|})$ is always true and I'm adding the same expression to both sides of the equation? What's going on here?

EDIT: Here's the WA output:

Answer 1

OK I tried it. Now what? Is this different from yours?