Why an exponential graph can't have b equal to 1

Question

I've seen that the graph of an exponential function, $f(x) = a\cdot b^x$, cannot have $b$ equal $1$. Why is this?

I think it's because the function would be a flat line if $b=1$. Is this true?

Answer 1

Yes, $1^x=1$ for all $x$. Hence if $b = 1$ we have that $$f(x) = a\cdot b^x = a\cdot 1^x = a\cdot 1 = a $$ is a straight horizontal line of height $a$.