Why do logarithmic curves not follow the general rules for transformations?

Elaborating on @Dave's comment, here's the sketch of what is going on.

• You know that if you have a function $f(x)$, then $f(-x)$ reflects the function over the $y$-axis.

• You know that if you have a function $f(x)$, then $f(x+3)$ shifts the function by $3$ to the left.

Now, consider your function:

• Starting with $f(x)=\ln(x)$, you look at $f(-x)=\ln(-x)$, which reflects the function over the $y$-axis.

• Now, let's call the new function we're dealing with $g$. In other words, $g(x)=\ln(-x)$. We would like to shift $3$ to the left, so we look at $g(x+3)=\ln(-(x+3))=\ln(-x-3)$.

• If you want to shift to the right by $3$, we could apply the change $g(x-3)=\ln(-(x-3))=\ln(3-x)$, which is what you get above.

The trick is that when you substitute $x+3$, you can't put the "$+3$" in anywhere, you have to replace $x$ by $x+3$ wherever it appears and remember to distribute. Therefore, in your case, the negative in front of the $x$ must be distributed to the $+3$ as well.