New Topic Exponents Exponential Equations I Exponential Functions The Exponential Function Base e The Logarithm Function Logarithmic & Exponential Equations Applications:Growth & Decay Review&Test

UNIT 5  : EXPONENTIAL & LOGARITHMIC FUNCTIONS

LESSON 5: THE LOGARITHMIC FUNCTION

Logarithmic Notation:

The exponential statement  23 = 8  may be rewritten in what is called logarithmic form

as follows:   log28 = 3  which reads:  the logarithm of 8, base 2, equals 3.

Any exponential statement may be rewritten in this form.

Example 1:

### Complete the following chart  by converting between exponential and logarithmic form.

 Exponential Form Logarithmic Form 34 = 81 52 = 25 log232 = 5 log101000 = 3

Solution:

 Exponential Form Logarithmic Form 34 = 81 log381 = 4 52 = 25 log525 = 2 25 = 32 log232 = 5 103 = 1000 log101000 = 3

Note:  When we read  log28, we ask the question  To what exponent must base 2 be raised to give 8?  The answer

is of course 3 and this idea gives rise to the following definition.

Definition:  the expression  logax  is defined to mean   the exponent to which base a must be raised

to give x.  The expression reads:   the logarithm of x, base a.

Hence  log10100 means the exponent to which base 10 must be raised to give 100.  The answer is 2, giving the

statement  log10100 = 2.

Hence  log381 means the exponent to which base 3 must be raised to give 81.  The answer is 4, giving the

statement  log381 = 4.

The Logarithmic Function:

Logarithmic functions are the inverses of exponential functions.  How does this come about??  See example below.

Table of Values & Graphs:

 x y -3 -2 -1 0 1 1 2 3 8 4 16

Now form the inverse of this function by interchanging x and y in the ordered pairs and the equation.

 x y -3 -2 -1 1 0 1 2 8 3 16 4

The equation of the inverse is found by interchanging x and y in the equation  y = 2x.  This yields  x = 2y.  If we put this

equation in logarithmic notation,   we obtain the logarithmic function   y = log2x.

The graph is a reflection in the line y = x of the exponential function y = 2x and is shown in red above.

Table of Values & graph:

 x y -3 -2 -1 3 0 1 1 2 3

Now form the inverse of this function by interchanging x and y in the ordered pairs and the equation.

 x y -3 -2 3 -1 1 0 1 2 3

The equation of the inverse is found by interchanging x and y in the equation  y = (1/3)x.  This yields  x = (1/3)y.  If we put this

equation in logarithmic notation,  we obtain the logarithmic function   y = log(1/3)x.

The graph is a reflection in the line y = x of the exponential function y = (1/3)x and is shown in red above.

The Natural Logarithmic Function  y = logex  or  y = ln x:

 x -3 -2 -1 0 1 1 2 3 4

Now form the inverse of this function by interchanging x and y in the ordered pairs above.

 x y -3 -2 -1 1 0 1 2 5 4

The equation of the inverse is found by interchanging x and y in the equation  y = ex.  This yields  x = ey.  If we put this

equation in logarithmic notation,  we obtain the logarithmic function   y = logex or y = lnx.

The graph is a reflection in the line y = x of the exponential function y = ex and is shown in red above.