This question was asked in a test and the use of calculator was not allowed.
Choose the correct one:
(A) $\log_e x$ can be defined as a real-valued function of $x$ for all $x\in R$
(B) $\log_{10}5$ is a rational number
(C) $\log_{10}5$ is an irrational number
(D) $\log_e x$ is algebraic number
I know that (A) cannot be the answer because $\log_e x$ is not defined for negative values of $x$
How do I check the validity of the other options without using a calculator. Please help.
Assume by contrary that $\log_{10}5\in \Bbb Q$ then we have$$\log_{10}5={p\over q}\quad,\quad \gcd(p,q)=1$$with $0<p<q$ since $1<5<10$, therefore$$5=10^{p\over q}\implies 5^q=10^p\implies 5^{q-p}=2^p$$which is impossible since no power of $2$ can be any power on $5$ except $1=2^0=5^0$. Finally we conclude that $\log_{10}5\notin \Bbb Q$ and (C) is correct.
Also (D) is wrong. Let $x=e^\pi$ and note that $\pi$ is transcendental.