Properties of Exponents: To review some of the fundamental properties of exponents, we will look at several examples.

If a > 0, b > 0, and x and y are any real numbers, then:

1. a^x ● a^y = a^x+y

 

 

2. 

 

3. 

 

 

4. 

 

5. 

 

 

Properties of Logarithms: To review some of the fundamental properties of logarithms, we will look at several examples.

If x and y are both positive numbers, then:

1. Meaning of Log: If .

 

If

Notice: the log is the exponent.

 

2. 

 

log(x) + log(x + 2) = log(x² + 2x)

 

Notice: if the base b is not shown, the base is assumed to be 10.

 

3. 

 

 

4. 

 

 

5. ln(x) means log_e(x) where .

 

 

6. 

 

 

Putting It All Together:

Solve for x if .

 

By direction inspection we can see that x must equal 4.

 

Solve for x if .

 

In this case, direct inspection fails us since 4 is too big - giving us 16 - and 3 is too small - only giving us 8.

 

To get an exact answer, we would use logarithms:

 

Step 1:

Step 2:

Step 3:

 

Let's Practice. In the following five equations, solve for x (accurate to 3 decimal places).

Graphs: A graph of both is shown below right. Notice that in order to graph , it is necessary to rewrite as since there is no log base 2 button on the calculator. Also, notice that the point (1, 2) is on . Since is its inverse, the point (2, 1) will be on that graph.

  

Derivatives: Given below are ten examples of taking derivatives of exponential functions and logarithms.

1. 
   
   
2. 
   
   
3. 
   
   
4. 
   
   
5. 
   
   
6. 
   
   
7. 
   
   
8. 
   
   
9.  
   
   
10. Show that has a relative minimum at .
    
    

We know from example #3 above that .

 

If we set this equal to 0, we get .

 

We note that for , this derivative is negative, and for , the derivative is positive.

 

This change in sign from negative to positive is the proof that we need to know that has a minimum value when .

 

 

      
Let's Practice.