What is the value of $1^x$? I am trying to understand why $1^{x}=1$ for any $x\in\mathbb{R}$
Is it OK to write $1^{x}$? As the base 1 should not equal 1 for $1^{x}$ to be an exponential function? 
Is $1^{x}=1$ just because it is defined to be so?
If possible please refer me to a book or article that discusses this topic.

I think you will at least agree that $1^x$=$1$ if $x$ is any natural number (since this is just 1 times itself x times).  We can extend this to all integer values of $x$ by using the facts $a^0$=$1$ for all non-zero $a$ and $x^{-c}$=$1\over{x^c}$.  Then we have $1^x$=1 for all rational numbers $x$ by the fact $1^{b\over{c}}$ is the cth root of $1^b$ which is 1 by previous results.  Finally we exdend this to all real values of $x$ by using the following definition of the exponential function from calculus.
"If $t$ is irational, then $a^t$ is defined to be the limit as n approaches infinity of $a^{t_n}$ where {$t_n$} is any sequence of rational numbers converging to $t$ (assuming this limit exists)."
In the case of $1^t$ we already have that $1^{t_n}=1$ for all rational $t_n$ so any such sequence as described above is the constant sequence, 1,1,1,..... which converges to 1.
