1. Exponential Functions

Exponential Function: f(x) = ax for every real x where a is positive and not equal to one.

Laws of Exponents.

The base e: Approximate . This is the natural exponential function.

 

Graphs of exponential functions:

1. Note: This is form for all bases less than 1.

This form known as exponential decay.

 

 

 

2. Note: This is the form for all bases greater than 1.

This form is known as exponential growth.

 

Reminder of graphing transformations:

From

Graph

 

 

Theorem: Exponential functions are one-to-one. (If the bases are equal and the exponents are equal, the bases to the exponents are equal.) This helps in solving equations.

Examples:

    1. Logarithmic Functions

      2. The inverse function of an exponential function is a logarithmic function.

      a is the base.

      When a = e, we call it the natural logarithm and abbreviate it "ln".

      When a = 10, we call it the common logarithm and do not use a base.

       

      Equivalent forms

      Logarithmic form

      Exponential form

      log464 = 3

      43 = 64

      log749 = 2

      72 = 49

      3a4t = 10

      log 1,000 = 3

      103 = 1000

      ln 1 = 0

      e0 = 1

      Finding logarithms: The logarithm of a number asks, "The base to what power equals this number?"

      Examples:

      log8 1

      8x = 1

      x = 0

      log9 9

      9x = 9

      x = 1

      log5 0

      5x = 0

      Not possible

      log6 67

      6x = 67

      x=7

       

      Graphing Logarithmic Functions: Note the domain.

      f(x) = log3x

      x

      y

      1

      0

      3

      1

      9

      2

      Logarithmic Functions are one-to-one, so we can easily solve equations.

      Examples:

      log7x = 3

      e-2x+1 = 13

    2. Properties of Logarithms

      3. Remember that the logarithm is simply a method of bringing the exponents to the base. So rules for exponents have analogous rules for logarithms:

       

      Exponent

      Logarithm

      Multiplication rule

      Division rule

      Power rule

      Addition rule

         

      logarithms are not distributive!

      We use these rules to expand and condense logarithmic expressions:

      Change of Base Formula:

      Examples of the usefulness of this formula:

    3. Logarithmic and Exponential Equations

These rules help us solve equations:

Logarithmic

1) Write both sides as one logarithm (with the same base).

2) Set the log values equal to each other.

3) Solve the polynomial equation.

4) Check the domain; you cannot take the logarithm of a negative number!

Examples:

 

Exponential:

  1. Get the base alone.

Take the log of both sides.

Solve.

Working with e: Follow the same rules as other algebraic manipulations.

 

4.6 Compound Interest

Compound Interest Formula:

Example:

$10,000 at 12% interest, compounded monthly

 

When interest is compounded continuously, the interest formula becomes:

Example: $100 @12.5% interest for 10 years

How much was invested @9.5% interest for four years, if the final balance is $15,000?
    1. Growth and Decay; Newton’s Law; Logistic Models

Exponential form of functions:

Example:

Exponential Growth/Decay Formula: