the natural base example 1 simplify natural base expressions simplify the expression. a.e2e2 e5e5 =...
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![Page 1: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/1.jpg)
7.3 FUNCTIONS INVOLVING
![Page 2: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/2.jpg)
The history of mathematics is marked by the discovery of special numbers such as and Another
special number is denoted by the letter e. The
number is called the natural base e or the Euler number after is discoverer, Leonhard Euler.
THE NATURAL BASE 𝑒𝑥
![Page 3: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/3.jpg)
EXAMPLE 1 Simplify natural base expressions
Simplify the expression.
a. e2 e5= e2 + 5
= e7
b. 12e4
3e3 = e4 – 34
= 4e
(5 )c. e –3x 2 = 52(e –3x)2
= 25e –6x
= 25e6x
![Page 4: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/4.jpg)
EXAMPLE 2 Evaluate natural base expressions
Use a calculator to evaluate the expression.
a. e4
b. e –0.09
Expression Keystrokes Display
54.59815003
0.9139311853
[ ]4
0.09[ ]
![Page 5: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/5.jpg)
GUIDED PRACTICE for Examples 1 and 2
Simplify the expression.
1. e7 e4
e11
3. 24e8
4e5
12e3
2. 2e–3 6e5
6e3SOLUTION
SOLUTION
SOLUTION
![Page 6: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/6.jpg)
GUIDED PRACTICE for Examples 1 and 2
Simplify the expression.
4. (10e –4x)3
1000e12x
Use a calculator to evaluate5. e 3/4
2.117SOLUTION
SOLUTION
![Page 7: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/7.jpg)
EXAMPLE 3 Graph natural base functions
Graph the function. State the domain and range.
a. y =3e 0.25x
SOLUTION
Because a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plot the points (0, 3) and (1, 3.85) and draw the curve.
The domain is all real numbers, and the range is y > 0.
![Page 8: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/8.jpg)
EXAMPLE 3 Graph natural base functions
Graph the function. State the domain and range.
SOLUTION
The domain is all real numbers, and the range is y > 1.
b. y = e –0.75(x – 2) + 1
a = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit.
e –0.75x
![Page 9: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/9.jpg)
CONTINUOUSLY COMPOUNDED INTEREST
![Page 10: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/10.jpg)
EXAMPLE 5 Model continuously compounded interest
A = Pert
SOLUTION
Finance
You deposit $4000 in an account that pays 6% annual interest compounded continuously. What is the balance after 1 year?
Use the formula for continuously compounded interest.
Write formula.
Substitute 4000 for P, 0.06 for r, and 1 for t.= 4000e0.06(1)
4247.35 Use a calculator.
The balance at the end of 1 year is $4247.35ANSWER
![Page 11: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/11.jpg)
GUIDED PRACTICE for Example 5
10. FINANCE: You deposit $2500 in an account that pays 5% annual interest compounded continuously. Find the balance after each amount of time?
The balance at the end of 2 years is $2762.93.ANSWER
a. 2 years
The balance at the end of 5 years is $3210.06.ANSWER
b. 5 years
The balance at the end of 7.5 year is $3637.48.ANSWER
c. 7.5 years
![Page 12: THE NATURAL BASE EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e 2 + 5 = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e](https://reader035.vdocument.in/reader035/viewer/2022070410/56649f0b5503460f94c1ecfe/html5/thumbnails/12.jpg)
GUIDED PRACTICE for Example 5
11. FINANCE: Find the amount of interest earned in parts (a) – (c) of Exercise 10.
A =a. 2500
$262.93ANSWER
A =b. 2500
$710.06ANSWER
A =c. 2500
$1137.48ANSWER