Properties of the Exponential functions
For x and y real numbers:
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axay = ax + y
example: 2325 = 28
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(ax)y = axy
example: (42)5 = 410
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(ab)x = axbx
example: (3*7)3 = 3373
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(a/b)x = ax/bx
example: (3/5)3 = 33/53
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ax/ay = ax - y
example: 57/54 = 53
Example 1 : Simplify the following expression
2x - 2x + 1
Solution to Example 1:
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Use property (1) above to write the term 2x + 1 as 2x2 in the given expression
2x - 2x + 1 = 2x - 2x2
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Factor 2x out
2x - 2x + 1 = 2x(1 - 2)
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Simplify to obtain
2x - 2x + 1 = -2x
Matched Exercise 1: Simplify the following expression
3x - 3x + 1
Example 2 : Find parameters A and k so that f(1) = 1 and f(2) = 2, where f is an exponential function given by
f(x) = Aekx
Solution to Example 2:
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Use the fact that f(1) = 1 to obtain
1 = Aek
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Now use f(2) = 2 to obtain
2 = Ae2k
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Multiply all terms of the equation obtained in step 1 by -2
-2 = -2Aek
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Add the equation in steps 2 and 3
2 - 2 = Ae2k - 2Aek
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and simplify
Ae2k - 2Aek = 0
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Factor Aek out.
Aek(ek - 2) = 0
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Neither A nor ek can be equal to zero. Therefore
(ek - 2) = 0
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Rewrite the above equation as follows
ek = 2
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Take the ln of both sides
k = ln(2)
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To obtain parameterA, substitute the value of k obtained in the equation obtained in step 1.
1 = Aeln(2)
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Simplify and solve for A.
A = 1/2
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Function f is given by
f(x) = (1/2)exln(2)
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Which can be written as
f(x) = (1/2)(eln(2))x
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and simplified to
f(x) = 2x - 1
Check answer
f(1) = 21 - 1
= 1
f(2) = 22 - 1
= 2
Matched Exercise 2: Find parameters A and k so that f(1) = 3 and f(2) = 9, where f is an exponential function given by
f(x) = Aekx
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