Exponential functions are those that have a variable as their exponent. This is something we have not seen up to this point in the other families.

The graphs of exponential functions are readily identifiable in that they gradually approach (but never touch) a boundary line (asymptote).

The base "b" is a constant number, which signifies either growth or decay.

• If b > 1, then we will see exponential growth

• If 0 < b < 1, then we will see exponential decay

Exponential functions can be transformed using the same transformation techniques we learned in the other function families. In addition to being a vertical stretch/compression value, "a" stands for the initial amount of something that will either grow or decay.

The exponent in an exponential function has a special name: logarithm. In order to solve exponential equations we need to "undo" the exponential expression and bring the variable (logarithm) down to ground level. And so, the inverse of an exponential function is called a logarithmic function. We will learn how to convert back and forth between exponential and logarithmic forms.

The bases we will deal with mostly in this course are 2, 10, and e. A logarithm with base 10 is so common that it is called the "common logarithm" and is usually not even written. The natural base e is approximately 2.7 and is special because it represents the kind of continuous growth or decay often found in the natural world.

When dealing with the natural base e, the inverse of this exponential function would be the natural logarithm, written as ln.

Applications of exponential functions often include:

• Biological/population growth

• Radioactive decay

• Growth of invested money (e.g., compounded interest)