3.4 Solving Exponential and Logarithmic Equations

3.4 Solving Exponential and Logarithmic Equations

Exponential or logarithmic functions can be solve by the One-to-One Properties or the Key to Everything.

One-to-One Properties

 if and only if 

 if and only if  

The Key to Everything

Examples:

                        One-to-One

  

                                      

                                                   The Key to Everything

                                           

     

        

                   

 Imaginary

 Continuous Compound Interest

A = balance

P = principle

r = annual interest rate in decimal

t = time over years

Examples:
                              If P = 10000, r = 0.0225, and let money to be triple. t = ?

    

3.3 Properties of Logarithms

 Properties of Logarithms

Logarithms possess certain properties that make them far more manageable in an algebraic setting than they would otherwise seem.  These properties allow one to manipulate them, just as a puppeteer might command a marionette.

The basic properties consist of the following:

• Assuming A is a positive number such that A is not equal to 0, N is a real number, and U and V are positive real numbers then

The first two properties are derived from the multiplicative and divisional properties of exponents since Logarithms are the inverses of exponential functions.  Multiplying two of the same base with exponents will result in that base raised to the sum of the exponents:

This rule for the exponents applies the same way to the terms within logarithms:

The rule is very similar for division:

Therefore:

The third property can now be explained using the other two properties.  If:

Then:

In this way it is always possible to remove an exponent from within a logarithm by moving it out in front of the logarithm.  The exponent that is being moved must apply to everything within the
logarithm.  It cannot just be the exponent of one of the factors or terms.  It must be an exponent that all factors and terms are raised to.

This is correct:

This on the other hand is an abomination:

Change of Base Formula

The change of base formula is as follows:

 

Where  a, b and x are positive real numbers such that a does not equal 1 and b does not equal 1.

 

The change of base formula can be used to rewrite a logarithm in a form that uses a base that you choose, b.  In this way strange bases can be converted to 10 or e so that you can use your handy dandy calculator to find the value of the logarithm.

 

This formula can be derived using the basic properties of a logarithm.

 

Let's evaluate:

Let's assign the variable y as equal to the logarithm:

 

Now it can be rearranged using the various properties of logarithms.  First use the basic principle of what it is to be a logarithm to rewrite it.

 

Then use algebra and the three properties of logarithms to rearrange it.  First take the log (of any desired base) of both sides:

Next, use algebra and the property of logarithms to isolate y.

Ian Hoeck

3.2 Logarithmic Functions and their Graphs

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs

Exponential functions are defined as functions with a base raised to an exponent x, were x is the unknown.

Although, there are certain restrictions on the variable a (which is called the base).

                                         These restrictions are and

                                          Or known as

The domain of an exponential function is 

The range of an exponential function is

To explain these further,  because if it did, you would have the graph y=1

If a were to be a negative number, you wouldn't have much of a graph, so you cant have a be negative.

Some basic things that would be good to know are:

Another basic rule to know is that anything raised to the 0 power will equal one.

The parent function known as  has some basic rules.

The y-int. will always be 1, because when x=0 (as said above), y=1.

The horizontal asymptote will be x=0.

And there is no vertical asymptote.

But of course you won't always be dealing with the basic parent function. Here are the modifications you could make to it.

By changing a, b, c and d, you can make the graph either compress, stretch, move left right or up and down.

First off, if b is less than 0 the function is known to decay. For example a graph of

Notice how the larger b gets the smaller y will be.

Now if   then the graph would look like this. And the larger b gets the steeper the graph would be.                                                            

                                                               

                                 

Notice how the bigger b is the steeper the graph becomes as you increase x.

The variable a determines the vertical stretch or compress that the graph will undergo.

The variable c determines the left or right shift that the graph undergoes (+ to x moves left, while - moves right)

The variable d determines the up or down shift of the graph (if d>0 then the graph will shift up d units, if d<0 the graph will shift down d units).


For some longer explanations on graphing or solving.