Why exp(x) is defined rather than proved in section 6.14 of Tom Apostol's calculus Vol 1

Solution 1:

My question is why did he decide to define $e^x$ for all real numbers, rather than prove it from the definition of the logarithm he gave in section $6.3$ and the number $e$ he gave in section $6.5$?

Because at that point, $e^x$ is not defined for irrational numbers, so the author had nothing to prove.

At the point when $e^x$ is defined, the author only defined $a^b$ for rational values of $b$, while the value is not defined for irrational values of $b$.

If I produce a new definition of $e^x$ and claim that this new definition is a definition for all real values, I need to verify that the new definition is consistent with existing definitions. That means I need to prove that, for rational values of $x$, the value $E(x)$ is equal to $e^x$, where $e^x$ is defined using roots and multiplications.

For irrational values, there hasn't yet been any definition of what $e^x$ is, so we are, at that point, completely free to define $e^x$ as whatever we want.

Looking at it from a different perspective, the answer to your question is really simple: It is because definitions are not something you prove.