Prove that$ f(x)=\ln(x)$, where $ x>0$ is of exponential order.

Question

Prove that$ f(x)=\ln(x)$, where $ x>0$ is of exponential order.

I know that if there exists a constant a and positive constants $t_0$ and $M$ such that $|f(t)| \leq M e^{at}$ at for all $t > t_0$ at which f(t) is defined then the function f is said to be of exponential order.

I also know that we are dealing with Laplace integrals in this section.

What I do not know is how to use this information to provide a proof.

Answer 1

Hint. You may easily prove $$ 0\leq\ln t \leq t \leq e^t,\qquad t\geq1 $$ giving $$|f(t)| \leq e^{t},\qquad t\geq1.$$