3.4 Solving Exponential and Logarithmic Equations

 By Mitch Boorstein
One-to-One Property


  if and only if x=y

        
 if and only if x=y


a>0
a does not equal 1

Example






x=5

Inverse Property 







Examples




x=6

OR











One to One Property in words:

If the bases of an exponential function are the same, then the equation above is true if x and y are the same.

Inverse Property in words:

You can raise both sides of a logarithmic equation to a certain power in order to cancel out the logs and solve the equation


Still Confused?
Here are some more examples
http://www.youtube.com/watch?v=tQfP3Twqeb4
http://www.youtube.com/watch?v=SuNKNnwH2-Q

3.3 Properties of Logarithms

Properties of Logarithms 

Changing Bases

Changing the base of a logarithm is very useful for solving logarithms on a calculator or . Most calculators only can solve common logs with a base of 10 or natural logs with a base e. 

The equation  

 

is used to change the base of a log.

In the equation c is the new base and a is the old base. 

Ex.1

In this case the base of 3 was changed to base 10.

This can then be solved on a calculator.

*Remember that the old base of 3 goes in the denominator and 243 goes in the numerator

Adding Logarithms 

Logarithms that share a same base and are being added together can be rewritten as  

.

Ex.

By separating the equations and solving them both, it is seen that they both end with the same solution.

                            

   

                  

                                 

Subtracting Logarithms 

Logarithms that share a same base and are being subtracted can also be written as 

.

However, 

This property only works one way.

Ex. 

       

Logarithms with Exponents 

Some logs have an exponent attached to it. Just as with addition and subtraction there is another way to rewrite the equation.

Ex.

3.1 Exponential Functions

 

 

 

3.1 Exponential Functions
The exponential function f with base a is denoted using

                
                      

  where , , and x is any real number

* a = 1 excluded because it yields f(x)=1 which is a constant

The graph of  is: 

Transformation of Graphs of Exponential Functions

Adding or subtracting from f(x) - the y value - shifts the graph of an exponential function up or down respectively:

     Examples:

• Adding two to shifts the graph up two units

                      

• The opposite is also true when subtracting two from f(x)= in that it will shift the graph down two units
•  

Conversely, adding or subtracting from the x value of the function shifts the graph left and right irrespectively

 

• Adding two to the x value of will shift the graph left two units 

• The opposite is also true when subtracting two from the x value of in that it shifts the graph right two units 

                           

  can also be reflected across the x and y axes when f(x)>0 and when x<0 respectively

                                                             

 

 

 NOTE - Be sure to examine when the y-intercept changes - or doesn't change - during each transformation

 

Compound Interest

Continuously Compounded Interest

P = Principal

R = Annual Interest Rate

N = Time

 

To account for more frequent (quarterly, monthly, daily) compounding of interest

 

n = Number of compounding's per year

t = Number of years

nt = total number of compoundings

 

Interest rate per compounding period is r/n, and the account balance after t years is:
 

 

                                                   

 

Examples of things that experience exponential growth or exponential decay are:

1. Population
2. Radioactivity
3. Interest 

 

2.6 Asymptotes and Intercepts

2.6 Asymptotes and Intercepts

Definition of Vertical and Horizontal Asymptotes

Vertical- The line x=a is a vertical asymptote of the graph of f if f(x) approaches infinity or f(x) approaches negative infinity as x approaches a, either from the right or left
Horizontal- The line y=b is a horizontal asymptote of the graph of f if f(x) approaches b as x approaches infinity or x approaches negative infinity.  
                                                         

               

The function above has 2 vertical asymptotes and one horizontal asymptote

vertical: x=1 and x=-1

horizontal: y=1

Rational Function example:

The vertical asymptotes come from the zeroes of the denominator

• Two vertical asymptotes are at +-3/2
• Since the degrees of the numerator and denominator are the same then this rational has a non zero horizontal asymptote
• The horizontal asymptote is found by dividing the leading terms

• Horizontal asymptote y=1/4

Example 2:

• y=2/3 is the horizontal asymptote

• If the exponent in the denominator is larger than the exponent in the numerator then the horizontal asymptote will be y=0
• As x approaches positive or negative infinity, the denominator will be much larger than the numerator thus making the overall fraction equal zero.

Example 3: 

• A function that isn't rational may have 2 horizontal asymptotes 
• one to the left and one to the right