**Key Terms**

**Objectives**

*e*

*Exponential Functions*

You should already be familiar with exponents. By way of review, however, here are the basic rules of math involving exponents. Note first that in the expression *a ^{b},*

*a*is the

**base**and

*b*is the

**exponent.**The rules below are expressed in terms of the base

*e,*which is a special irrational number with a variety of applications in math and science. The number

*e*is approximately equal to 2.71828. These rules apply to any base, however.

_{} _{} _{}

Let's now look at the simple exponential function _{}, which is plotted below.

This function has a domain (-∞, ∞) and a range (0, ∞). It also has an asymptote at *y* = 0 (the *x-*axis); note that 0 is not in the range of the function. Furthermore, it intercepts the *y-*axis at *f*(0) = 1 (since any number raised to the zeroth power is 1), but it has no real roots.

This function has application, for instance, in the case of interest on investments. Given a principle investment *P* and a continuously compounded interest rate *r,* the total value *V* of the investment at time *t* (where *t* and *r* are both expressed in terms of the same unit of time) is

_{}

Note in this case that the *y*-intercept (value at *t* = 0) is *V*(0) = *P,* or the initial investment amount. The value of the investment grows at an increasing rate as time goes on (hence the fact that a small interest rate can greatly increase value in a fairly short amount of time). Below is a plot of the value *V*(*t*) of a $1,000 investment with an annual interest rate of 5%. Here, *t* is measured in years. The function is _{}. After 20 years, even at just 5% interest, the initial investment has nearly tripled.

Also noteworthy is the exponential function _{}, which is plotted below. Note that unlike _{}, where the function increases as *x* increases, the function _{} decreases as *x* increases. Functions similar to this one are useful for modeling physical phenomenon that involve decay over time, such as the decreasing amplitude of a spring in motion as friction works on it. The function _{} has the same domain and range as _{}.

*Logarithmic Functions*

A **logarithm** is the inverse function of exponentiation. Let's say we have a function _{}. By our definition of inverse functions, a logarithmic function *g*(*x*) (the inverse of *f*(*x*)) would satisfy the following expression.

_{}

Generally, the simple logarithmic function has the following form, where *a* is the base of the logarithm (corresponding, not coincidentally, to the base of the exponential function).

_{}

When the base *a* is equal to *e,* the logarithm has a special name: the **natural logarithm,** which we write as ln *x.* This natural logarithmic function is the inverse of the exponential _{}. Thus,

_{}

This means that the following two equations must both be true.

_{} _{}

Below are the basic rules of logarithms. These are expressed generally using the arbitrary base *a, *but they apply when *a* = *e* and the logarithm is expressed as ln (which is identical to log* _{e}*).

_{} _{} _{} _{} _{} _{}

Below is a graph of the natural logarithm function. Note that because the exponential _{} is always positive for real values of *x,* the domain of the function ln *x* is (0, ∞). A vertical asymptote exists at *x* = 0. The range of the function is (-∞, ∞). You might also notice that the graph of the function ln* x* looks like the graph of the function _{} rotated clockwise 90° and then rotated 180° around the vertical axis. This is a general characteristic of inverse functions.

__Practice Problem:__ How long does it take for an initial investment of $100 to double, given an annual interest rate of 10% that is compounded continuously?

__Solution:__ We can use the rules of exponents and logarithms to solve this problem. Recall from above that _{}, where *P* is the initial investment (principal), *r* is the interest rate, and *V* is the value of the investment at time *t* (expressed in years)*.* When $100 has doubled, it is $200:

_{}

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We can apply natural logarithms to solve this problem. Take the natural logarithm of both sides and apply the rules of logarithms (we drop the dollar signs for simplicity):

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Thus, at a continuously compounded annual interest rate of 10%, an investment doubles roughly every seven years.

__Practice Problem:__ Solve the equation below for the variable *c.*

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__Solution:__ Here, we can apply the rule of inverse functions:

_{}

Thus, apply the exponential function to both sides of the given equation, then evaluate each side.

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Now, check the result by plugging it back into the original equation.

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