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 8: LOGARITHMIC EQUATIONS SUMMARY AND REVIEW

LESSON 1:  LAWS OF EXPONENTS

Examples of Powers:

25 = 2 x 2 x 2 x 2 x 2 = 32;   (-3)3 = (-3) x (-3) x (-3) = -27;    2.72 = 2.7 x 2.7 = 7.29

A POWER  (am ) consists of two parts; the base a and the exponent m.

# Review of Basic Exponent Laws:

 Rule Example Explanation am x an = am+n 32 x 35 = 37 Multiplication Rule - If the bases are the same, add the exponents Division Rule - If the bases are the same, subtract the exponents (am)n = amn (32)3=36 Power Rule  When taking a power of a power, multiply the exponents (ab)m = amam (3 x 2)4 = 34 x 24 Power of a Product  Take each factor in the product to that power Power of a Quotient  Take numerator and denominator to that power

Zero Exponents:     Rule:     a0 = 1

Examples :  20 = 1;     (-3.4)0 = 1   (x2)0 = 1;            Note that  00 is not defined.

Examples :

Example :  Simplify

Solutions:

Rational Exponents:

Examples :

LESSON 2:  Exponential Equations:  2x = 64;    92x  1 = 729;      500(1.02)x  1 = 897.56

Theorem:  If  ax = ay, then x = y.    In words:  If an exponential equation has the bases equal, then the exponents must be equal.

This theorem gives us our strategy for solving exponential equations, namely convert each side of the equation to a common base.

### Growth and Decay problems:

Example:

The half-life of radium is 1600 years.  What fraction of radium remains from a sample after 12 800 years?

Solution: Again let the starting mass be 1 g.

Recall:

 A  = ? A0 = 1 g    t = 12 800 years   h = 1 600 years

·        A is the mass remaining after the decay period

·        A0 is the original mass of radioactive material

·        ½ is the decay factor

·        t is the total time elapsed

·        h is the half-life of the material

Example:

A bacteria culture doubles in size every 20 min.  How long will it take for a sample of 10 bacteria to grow to 20 480?

Solution:

 A  = 20 480 A0 = 10    t = ?   d = 20 min.

Therefore it will take 3 2/3 hours.

LESSON 3:  Exponential Functions:

LESSON 4: THE EXPONENTIAL FUNCTION  y = ex

Table of Values & graph:

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

LESSON 5:  The Logarithmic Functions:

Logarithmic functions are the inverses of exponential functions.

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.

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

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

Now form the inverse  interchange the components of the ordered pairs:

 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.

Logarithms:

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.

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

The following properties of logarithms are important and used frequently in our study of logarithms.  They correspond closely to

our rules for exponents studied earlier.

The following properties of natural logarithms are important and used frequently in our study of logarithms.

They correspond closely to our previous rules for logarithms studied above.

LESSON 6:  LOGARITHMIC EQUATIONS

### LESSON 7:  Growth and Decay Problems Revisited:

Growth and decay problems are governed by the following formula: