How would you interpret this equation?

Question

$$ y = \mathrm{ln}~3x(x - 3)^2 $$ Solving the derivative I saw it as $(\mathrm{ln}~3x)(x-3)^2$ and used the product rule but I was told by my professor I should have solved it by taking the entire term under the natural log as so $\mathrm{ln}[3x(x-3)^2]$ Was I wrong interpreting it the way I did?

Answer 1

I agree that the expression might be considered somewhat ambiguous and that parentheses could have been of help, for example writing $\ln \left(3x(x-3)^2\right)$, but I would also have interpreted $\ln 3x(x-3)^2$ in the way that your professor did.

For example, consider the basic property of logarithms $\ln ab=\ln a+\ln b$. Notice that we naturally interpret $\ln ab$ to mean $\ln (ab)$, and not $(\ln a)b$. In your example, we simply have $a=3x$, $b=(x-3)^2$. To mean $(\ln a)b$, we would have written $b\ln a$.

In the same sense, if we had meant the expression in the way that you interpreted it, we would most likely have written $(x-3)^2\ln 3x$.