What does really mean by- power of a number or an exponential function is dimensionless? Is power of only a number or an exponential function is dimensionless? If power of any other thing can also be dimensionless then please explain with examples.
 A: To consider an example, take the case of exponential decay
$$N=N_\circ e^{-\lambda t}$$
We can write this as 
\begin{eqnarray*}
N & = & \frac{N_{\circ}}{e^{\lambda t}}\\
 & = & \frac{N_{\circ}}{\underbrace{e\times e\times e\times e\times\ldots \times e}_{\lambda t\text{ times}}}
\end{eqnarray*}
So $\lambda t$ must be a dimensionless term that is telling how many times we should multiply $e$ by itself. Thus, $\lambda t$ must be dimensionless "overall". Individually, $\lambda$ has the dimensions of $[T^{-1}]$ which cancels with $t$ to give a net dimensionless quantity.
$\underbrace{e\times e\times e\times \ldots}_{10 \text{ meters times}}$ makes no sense mathematically.
We could have taken a dimensional quantity instead of $e$ but the exponent $\lambda t$ would still be dimensionless. eg in the kinematical equation $s=ut + \frac 12 at^2$,  $t^2$ has the dimensions of $[T^2]$ but the exponent $2$ is dimensionless.
The same applies to transcendental functions i.e. logarithmic, trigonometric, etc.
