Math Lesson 15.5.8 - Exponential Graphs that Involve Euler's Number

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Welcome to our Math lesson on Exponential Graphs that Involve Euler's Number, this is the eighth lesson of our suite of math lessons covering the topic of Exponential Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Exponential Graphs that Involve Euler's Number

The most famous exponential functions perhaps are those that have Euler's Number as a base. In such functions, we write the letter 'e' as a base instead of numbers such as 2, 3, 4, etc. In the previous chapter, we explained that Euler's Number e is an irrational number slightly greater than 2.7 (e ≈ 2.71828). In this way, the exponential graphs produced by such functions are similar to those explained at the beginning of this tutorial, which have a base greater than 1.

The parent function of exponential functions that include Euler's Number is

and the most frequent exponential function is

where k is the original y-value for x = 0.

The most typical situations involving Euler's Number as the base of exponential functions are those that show the number of elements in a sample as a function of time. For example, the number of bacteria left in a sample of a living organism as a function of the time elapsed since the start of the medication, the number of the non-decayed nuclei as a function of time elapsed in a radioactive sample, etc. Let's consider an example to clarify this point.

Example 7

The number of bacteria in a sample changes according to the exponential below

after starting the medication, N is the number of bacteria left in the sample and t is the time (in days) elapsed since the start of the medication.

1. Plot a Number of bacteria v's Time graph to show the relationship between these two quantities.
2. The bacteria become harmless if there are less than 5 bacteria in the sample. Use the graph to find the duration of the cure.

Solution 7

1. This situation is not different from those discussed earlier in this tutorial. The only modification to do is to label the horizontal axis as Time instead of X and the vertical axis as Number of bacteria instead of Y. Another thing to keep in mind here is that we don't need the negative part of the horizontal axis, as the time cannot be negative. From the clues, it is evident that there are 500 bacteria in the sample at the beginning of the cure, as for t = 0 we have
   N(0) = 500 ∙ e^{-0.75 ∙ 0}
   = 500 ∙ e⁰
   = 500 ∙ 1
   = 500 bacteria
   This means the highest point of the graph is 500 and this value decreases when moving to the right, as the coefficient m of this exponential function is negative (m = -0.75). We can find some other points needed to plot the graph by making a table as shown below. Now, we insert all these pairs of values into the coordinates system and connect them smoothly to plot the graph. The first five points of the table are not shown in the graph as they are in a higher position and the dimensions of the page are insufficient to show all points. The black dot shows the 6th point, which corresponds to the fifth day of the cure.
2. By looking the graph, it is easy to see that the number of days elapsed until there are 5 or less bacteria in the sample is slightly more than 6. Therefore, the patient has to take the cure for 7 days to heal completely.

You have reached the end of Math lesson 15.5.8 Exponential Graphs that Involve Euler's Number. There are 8 lessons in this physics tutorial covering Exponential Graphs, you can access all the lessons from this tutorial below.

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1. Euler Feedback. Helps other - Leave a rating for this euler (see below)
2. Types of Graphs Math tutorial: Exponential Graphs. Read the Exponential Graphs math tutorial and build your math knowledge of Types of Graphs
3. Types of Graphs Revision Notes: Exponential Graphs. Print the notes so you can revise the key points covered in the math tutorial for Exponential Graphs
4. Types of Graphs Practice Questions: Exponential Graphs. Test and improve your knowledge of Exponential Graphs with example questins and answers
5. Check your calculations for Types of Graphs questions with our excellent Types of Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Types of Graphs Calculators by iCalculator™ below.
6. Continuing learning types of graphs - read our next math tutorial: Circle Graphs

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