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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, you will learn what kind of number 'e' is. Also learn the formal name for this unique number. See what kind of graph the natural base 'e' graphs into, as well as the uses of these types of functions.

** e** is an interesting number.

For example, the phrase 'to express *e*, remember to memorize a sentence to simplify this' matches each word to each number. The beginning word has two letters matching it with the first number, 2. We place a dot after the two. Then the next word 'express' has seven letters matching it with the second number, 7. The following words continue in the same manner. If you want to memorize it and impress your friends, you can use this memory aid or any other memory aid that you know of.

When the number *e* is used as the base for a logarithm, we call the number *e* the natural base, and the logarithm is called a natural logarithm. Usually, logarithms are assumed to have a base of 10, so when we change the base to *e*, we write it as *ln(x)* or we write *log* with a subscript of *e* to show that it has *e* as the base instead of 10.

The way the natural log works is if we take the natural log of *x*, it will be equal to the power of *e* that gives you *x*. For example, the natural log of *e* is 1 since *e* to the first power is *e*. The natural log of 1, on the other hand, equals 0 because *e* has to be raised to the 0th power for it to equal 1.

The inverse function of the natural log of *x* is simply *e*^*x*. The inverse is the function that reverses the original function. So, this means that if *ln(x)* gives you *y*, then *e*^*y* will give you *x*. They reverse each other.

When graphed, the natural log of *x* curves and slowly approaches infinity as *x* increases. The function never touches the *y*-axis but goes to negative infinity the closer the function gets to *x* = 0.

The *y*-axis is therefore an asymptote of the natural log function.

This natural log function with its graph has quite an interesting use. For example, say you were told that you had a savings account where the continuous compound interest rate is five percent. You wouldn't be able to use this percentage in normal interest calculations since it is your continuous interest and not just a compound interest rate.

What does continuous mean? Continuous means that your interest is constantly being calculated and you would be paid every single moment that your interest is over a penny. But with the use of *e*, you can use the formula *e*^*r* - 1, where the *r* equals your continuous compound interest rate to calculate the compound interest rate that you can use in your other interest calculations.

What have we learned? We've learned that the number ** e** is an irrational number that begins with 2.718281828459045 and continues indefinitely with no pattern. The logarithm that uses the number

The graph of *ln*(*x*) increases slowly towards infinity as *x* increases. The *y*-axis is an asymptote where the function goes towards negative infinity as it approaches *x* = 0. The number *e* is used to calculate the compound interest rate given a continuous compound interest rate using the formula *e*^*r* - 1.

Viewing and studying this video lesson could strengthen your ability to:

- Understand the use of
*e*in the ln(x) and the inverse - Recognize a graph of ln(x)
- Apply
*e*to compounding calculations

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- What Is an Exponential Function? 7:24
- Exponential Growth vs. Decay 8:41
- Transformation of Exponential Functions: Examples & Summary 5:51
- Using the Natural Base e: Definition & Overview 4:47
- How to Evaluate Logarithms 6:45
- Writing the Inverse of Logarithmic Functions 7:09
- Exponentials, Logarithms & the Natural Log 8:36
- Basic Graphs & Shifted Graphs of Logarithmic Functions: Definition & Examples 8:08
- Logarithmic Properties 5:11
- Practice Problems for Logarithmic Properties 6:44
- How to Solve Logarithmic Equations 6:50
- Using the Change-of-Base Formula for Logarithms: Definition & Example 4:56
- How to Solve Exponential Equations 6:18
- Go to Exponential and Logarithmic Functions

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