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) = A e^{k x}
Solution to Example 2:

Use the fact that f(1) = 1 to obtain
1 = Ae^{k}

Now use f(2) = 2 to obtain
2 = A e^{2k}

Multiply all terms of the equation obtained in step 1 by 2
2 = 2 A e^{k}

Add the equation in steps 2 and 3
2  2 = A e^{2 k}  2 A e^{k}

and simplify
A e^{2 k}  2 A e^{k} = 0

Factor A e^{k} out.
A e^{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 = A e^{ln(2)}

Simplify and solve for A.
A = 1/2

Function f is given by
f(x) = (1/2) e^{ x ln(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) = A e^{ k x}
More References and Links Related to Exponential Functions