An exponential function raises a base to the power of x, so that as x increases, f(x) increases exponentially. In other words, x is in the exponent. and f(x) gets really big, and grows faster as x increases. The domain is all real numbers, since you can any number as an exponent, and the range is that f(x) must be greater than zero, because no number raised to a power is equal to zero.
In a logarithmic function, f(x) describes the power you have to raise a base to in order to get x. Since no number raised to a power is less than or equal to zero, the domain is x > 0. The range however, is technically all real numbers. Keep in mind though that x would have to be pretty darn big for you to have to raise the base to the thousandth power to get to it, so it looks like it has an asymptote within a certain boundary. Unlike exponential function, f(x) grows slower as x increases, although its slope is always positive.
