## Properties of the Exponential functions

For x and y real numbers:

- a
^{x}a^{y} = a^{x + y}

example: 2^{3}2^{5} = 2^{8}

- (a
^{x})^{y} = a^{xy}

example: (4^{2})^{5} = 4^{10}

- (ab)
^{x} = a^{x}b^{x}

example: (3*7)^{3} = 3^{3}7^{3}

- (a/b)
^{x} = a^{x}/b^{x}

example: (3/5)^{3} = 3^{3}/5^{3}

- a
^{x}/a^{y} = a^{x - y}

example: 5^{7}/5^{4} = 5^{3}

__Example 1 :__ Simplify the following expression

2^{x} - 2^{x + 1}

__Solution to Example 1:__

- Use property (1) above to write the term 2
^{x + 1} as 2^{x}2 in the given expression

2^{x} - 2^{x + 1} = 2^{x} - 2^{x}2

- Factor 2
^{x} out

2^{x} - 2^{x + 1} = 2^{x}(1 - 2)

- Simplify to obtain

2^{x} - 2^{x + 1} = -2^{x}

**Matched Exercise 1:** Simplify the following expression

3^{x} - 3^{x + 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) = Ae^{kx}
__Solution to Example 2:__

- Use the fact that f(1) = 1 to obtain

1 = Ae^{k}

- Now use f(2) = 2 to obtain

2 = Ae^{2k}

- Multiply all terms of the equation obtained in step 1 by -2

-2 = -2Ae^{k}

- Add the equation in steps 2 and 3

2 - 2 = Ae^{2k} - 2Ae^{k}

- and simplify

Ae^{2k} - 2Ae^{k} = 0

- Factor Ae
^{k} out.

Ae^{k}(e^{k} - 2) = 0

- Neither A nor e
^{k} can be equal to zero. Therefore

(e^{k} - 2) = 0

- Rewrite the above equation as follows

e^{k} = 2

- Take the ln of both sides

k = ln(2)

- To obtain parameterA, substitute the value of k obtained in the equation obtained in step 1.

1 = Ae^{ln(2)}

- Simplify and solve for A.

A = 1/2

- Function f is given by

f(x) = (1/2)e^{xln(2)}

- Which can be written as

f(x) = (1/2)(e^{ln(2)})^{x}

- and simplified to

f(x) = 2^{x - 1}

__Check answer__

f(1) = 2^{1 - 1}

= 1

f(2) = 2^{2 - 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) = Ae^{kx}

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