# Evaluating Exponentials & Logarithms on a Graphing Calculator

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

A graphing calculator is an advanced calculator that can evaluate simple and complex mathematical expressions including exponentials and logarithms. Let's look at the steps needed to evaluate these expressions.

## Exponentials and Logarithms

Suppose that a population of a certain probiotic is increasing at a rate such that it doubles each day. Based on this, you derive that after three days, the population of the probiotic is 23. This is an exponential expression.

An exponential is a mathematical expression of the form ab, where we call a the base and b the exponent. To evaluate these expressions, we multiply the base by itself the number of times that the exponent indicates. Therefore, to evaluate 23, we multiply 2 by itself 3 times.

• 23 = 2 × 2 × 2 = 8

The population after three days is equal to 8.

Now, suppose that we know the probiotic population after a certain number of days is 32, and we want to know how many days have passed to get to this. In other words, we want to know what power do we raise 2 to in order to get 32. This involves logarithms. Logarithms are mathematical expressions of the form:

loga (x)

We call a the base of the logarithm. To evaluate logarithms, we find the exponent we would raise a to in order to get x.

To find the number of days that have passed to get to a population of 32, we evaluate log2 (32). Notice,

• 2 × 2 × 2 × 2 × 2 = 32

Therefore, if we raise 2 to the exponent 5, we get 32, so log2 (32) = 5.

These are two simple examples, but what if we want to evaluate something like 318 or log2 (3)? Thankfully, we can use graphing calculators, which are calculators that can perform advanced operations and also have graphing capabilities.

Let's take a look at how to use these calculators to evaluate both exponentials and logarithms.

## Evaluating Exponentials

To evaluate the exponential expression ab, we make use of the exponent button on the calculator. This button shows up differently on different graphing calculators. Some common ones are

• ax
• ^
• Exp
• yx

To evaluate ab on a graphing calculator, we use the following keystrokes:

1. Enter a.
2. Hit the exponent button (^).
3. Enter b.
4. Hit enter.

Consider the expression we mentioned 318. To evaluate this, we use the following keystrokes:

1. 3
2. ^
3. 18
4. Enter

In doing so, we get that 318 = 387,420,489.

When numbers are very large or very small, the graphing calculator may give the answer in the form A E B, where A and B are numbers. The E indicates that we move the decimal point in A. We move it B units to the left if B is negative and B units to the right if B is positive.

For instance, to find the probiotic population after 34 days, we evaluate 234 on the graphing calculator and get 1.717986918 E 10. This says we move the decimal in 1.717986918 ten units to the right to get that there are 17,179,869,180 probiotic colonies after 34 days.

## Evaluating Logarithms

A graphing calculator can only evaluate logarithms with two specific bases; e and 10, where e is the irrational number with approximate value 2.71828. When we have a logarithm with base 10, we just write log(x), and when we have a logarithm of base e, we write ln(x). Base e logs are also called natural logs.

Because we can only evaluate logarithms with these two bases on a graphing calculator, we have to be able to change the base of any logarithm to one of these bases. To do this, we use the change of base formula:

For example, consider log2 (3). To evaluate this on a graphing calculator, we must first change to base 10 or base e (we can use either one). Let's use our change of base formula to change the base of log2 (3) to base e.

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