# Using Exponential & Logarithmic Functions to Solve Finance Problems

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Exponential and logarithmic functions can be seen in mathematical concepts in finance, specifically in compound interest. This relationship is illustrated by the exponential function and its natural logarithmic inverse.

## Continuously Compounding Interest

Exponential and logarithmic functions are used in several fields of study. They can be used to determine pH in chemistry or to show population growth in biology. One area of study that also implements exponential and logarithmic functions is finance. Specifically, compound interest can be calculated using these functions.

Let's say that we wish to invest a principal of one dollar (P = \$1) into some random investment, which pays a rate of return of 100% (or r = 1.00). Furthermore, this interest rate will be compounded continuously over the course of the year. How much money will we have at the end of one year? Two years? Four years?

Continually compounded interest is given in the general form:

A = Pe rt

• A is the total amount we are left with over a time
• P is the principle. For this problem P = \$1.
• r is the rate. In this case, r = 1.00, a return rate of 100%.
• t is time

If we were to graph this, time (t) is represented along the x-axis graphically, A is represented along the y-axis, and r is a constant. To find out how much we made on this investment we'd merely subtract P from A.

## Review of Natural Log

Before we get started, however, it may be helpful to expound on the concept of the natural log as it is seen in the exponential and logarithmic functions. To illustrate the inverse relationship between these two types of functions, we will look at the exponential function and its respective inverse logarithmic function with the natural log base e, also referred to as Euler's Number. (Note that e = 2.7182). Furthermore, we will see how these functions relate to continually compounded interest.

### Natural Log

The natural logarithmic function is defined as y = ln x, where e (2.7182) is merely a subscript of ln, denoting that it is a natural log function. This function y = ln x can be viewed graphically:

If we input some x-values into this function, our corresponding y-values appear as:

Note that there are no corresponding values for x < 0.

### Inverse of Natural Log

The expression y = ln x is equivalent to the exponential equation x = ey. Taking the inverse (switching the x and y) of x = ey, we get y = ex.

Therefore, the exponential function with the natural log base, y = e x, is the inverse of the natural log function y = ln x.

The equation y = e x behaves like other exponential functions graphically speaking. The y-values increase quickly with respect to x.

We see in the graph of y = ex:

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