ELF - Functions Involving Euler's Number e Lesson

Functions Involving Euler's Number e

In this lesson, we will explore the number e, an irrational number, approximately 2.71828183. The number e is named after Leonhard Euler, who was an 18^th century Swiss mathematician. The number e is a very famous number in the field of mathematics. e has great importance in mathematics; it is on the level of importance with the number equation image indicator .

We have already discussed and explored how exponential and logarithmic functions are inverses of each other. Now that we have been introduced to Euler's (pronounced "oyler") number e, we need to know that exponential functions with a base of e will be treated as exponential growth functions. As such, the same properties apply to functions with a base of e.

Logarithms that use the irrational number e as a base are of particular importance in many applications. The function y=log_ex is the natural logarithmic function and has a base of e. The shortened way y=log_ex is commonly represented is y=lnx.

Calculators are programmed to evaluate natural logarithms. Consider ln(34), which means the exponent to which the base e must be raised to obtain 34. The calculator evaluates ln(34) as approximately 3.526. This value makes sense because equation image indicator $LaTeX: e^{3.526}$ $e^{3.526}$ is approximately 33.9877, a value very close to 34. Also consider the problem: if LaTeX: e^x=85 $e^x=85$ , determine the value of x to the nearest hundredth. The hint for this problem is to write the exponential equation in logarithmic form. So, equation image indicator $e^x=85$ is equivalent to ln(85) = x, which is approximately 4.44.

Now let's explore functions involving Euler's number e. When applying the properties of exponents using e, we will treat e as if it were a variable, unless there is a practical application (word problem) or a need to graph using e.

image depicting two sets of equations:
The base number "e" uses the same properties as other exponential base numbers.

Watch the following videos that introduce us to functions involving Euler's number e, as well as some examples on how to simplify, evaluate, and graph these same functions.

It's now time for us to explore and practice working with functions involving Euler's number e.In the next lesson we will explore some more logarithmic properties, as well as see more of the application of functions involving Euler's number e.

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