Does $\ln x + f(x)$ mean $\ln(x) + f(x)$, or $\ln(x + f(x))$

Question

In an exercise from a math book there is a function $g$ defined by $\ln x + f(x)$, and what I don't understand is whether this means that $g$ is equal to the natural log of $x$, and then to the $\log$ you add $f(x)$, or is it equal to $\ln$ of $(x + f(x))$? In short, does $\ln x + f(x)$ mean: $\ln(x) + f(x)$ or $\ln(x + f(x))$ ?

Answer 1

In general, for any "non-made up" function like $\sin$, $\cos$, $\ln$, $\log$ etc. , if there are multiple monomials after the function, we only take the first monomial. For example, $\sin x+4x^2=(\sin x)+4x^2$, $\ln x+2=(\ln x)+2$ etc.

If there is parenthesis, then we only take everything inside the parathesis. For example, $\sin(x+4x^2)+5=[\sin(x+4x^2)]+5$.

In your case, since there are no parenthesis, it means $\ln (x)+f(x)$.

Answer 2

Without parentheses, it's usually assumed that the expression only extends to the immediate next symbol, so $\ln x$ is equal to $\ln(x)$.