What is an exponential function ?

The general form of the exponential function can be written :
               f(x) = ab^x
a is the initial value and b is the growth rate.

Let's look at this problem :
      The growth of bacteria in a dish is modeled by the function f(t) = 2^ n . For which value of t is f(t) = 32 ?
                                                                                                             3
The first step is to write out the exponential function .
          32 = 2^ t 
                       3
Next find a common base . To do this you can turn 32 into 2^5 because 2 to the fifth power is 32.

The new function becomes : 2^5 = 2^ t 
                                                          3
Now you solve for x : 5 =  t 
                                         3
You do this by multiplying 3 to every side.
You final answer becomes t = 15.


Now Try It Yourself : 
   A population of rabbits double every 60 days according to the formula P=10(2)^ 1 , where P is the population of rabbits on day t. 
                             60
What is the value of t when the population is 320 ? 

How Do We Solve Exponential Equations ?

Exponential Equations is when a variable is the exponent.
  For Example : 100 = 10^x  
        The missing exponent would be 2 because 10 squared is 100.

Let's try  2^11 = 2^x
Here's a rule to remember : When the Bases are the same, the exponents are equal.
So the missing exponent would be 11.

Lets try something harder.  3^x+4 = 3^7
Since the bases are the same, set up an equation with the exponents.
It looks like this : x+4=7
 You solve regularly for x, which becomes x=3.

Lets try one more problem . 2^x-3 = 4
First you have to create common bases. You can do this by turning 4 into 2^2.
Now the exponential equation becomes 2^x-3 = 2^2
You set up an equation : x-3 = 2, which becomes x=5.

Now try it Yourself : 2^3x+7 = 2^5x-1

How Do We Simplify Complex Fractions ?

Let's get       2       in simplest form
                    x     
                    1
                   2x

  The first step is to rewrite the problem as a division problem, so it would look like this :
                2    ÷   1 
                x        2x

  The next step is to divide as you would any fraction. Keep the first fraction, change the sign from division to multiplication and flip the second fraction.
It should look like this :

               2   *   2x 
               x         1

The next step is to multiply straight across.

                     4x  
                     1x
The final and last step is to cross out anything the numerator and denominator have in common which is X .

  Final Answer :    4   -------> 4 
                            1


Now Try It Yourself : 
                6  
               9x       
                5 
               3x