The author here discriminates between the two functions as a is a fixed number, and x is a variable. I am confused as to this distinction, as it seems the reason why we would need to use 'a' instead of a number is to account for many different possibilities of 'a', such that 'a' is essentially a variable. I would appreciate an explanation as to the difference between these two, and when something expressing the meaning of 'a' would be used as opposed to something with the meaning of 'x'. thanks
$\begingroup$ Both are of the form $a^b$. Saying one of $a,b$ is a "variable" means that you focus on the behavior as a function of that variable while considering the other one as a fixed parameter. If you call $a$ to be the variable the focus is on the power function $x^a$. If you call $b$ to be the variable the focus is on the exponential function $a^x$. $\endgroup$– dxivMar 29 at 5:17
It is not the "Distinction between Variable & Number" ; It is the "Distinction between Variable and FIXED Number" ....
In other words , we have Variables & Constants , both of which may be Numbers.
$f(x)=2^x$ , $g(x)=x^2$ , $h(x)=3x$ , $i(x)=3a^x$ are functions with 1 Variable , where $a$ is FIXED Constant.
$F(x,a)=a^x$ , $G(x,a)=x^a$ , $H(x,a)=ax$ , $I(x,a)=3a^x$ are functions with 2 Variables , where $a$ is not a Constant.
In the expressions$$a^x$$ and $$x^a,$$ $a$ is variously called a fixed number, a constant, an arbitrary constant and a parameter, to indicate that the value of $a$ can be arbitrarily fixed (so that we are considering an exponential function and a power function, respectively), after which we are considering the value of, say, $7^x$ or $x^7$ as $x$ varies. To capture how $a$ and $x$ vary, you can draw a family of curves, each curve corresponding to a value of $a,$ similar to how you might draw a family of curves to represent the antiderivatives $\int 2x\,\mathrm dx=x^2+C,$ where $C$ and $a$ play the same role.
In truth, the terminology for these placeholders is informal, and $a$ is indeed just a special variable.