Exponentials and Logarithms
Section 9.0A

Applications involving population, radioactive decay, carbon-dating, earthquakes and the decibel scale use exponential and logarithm properties.

Recall a Function is a correspondence between two sets; the first set are x values, and the second set are y values.

For each x value, there is exactly one y value.

Example 1:   

        y = 2x + 3

When 

        x = 1,     y = 5

        x = 0,     y = 3

x is the independent variable, y the dependent variable.

This function is a linear function, the equation is of a line, with constant slope = 2

        Slope =    rise     = change in y     = 5 – 3    = 2
                        Run        change in x       1 – 0

An exponential function is of the form:                    

        y = b^x   ; b > 0 ; b 1

  The base is b, and is constant.  The independent variable x, is the exponent.

Example 2: 

Sketch             

        y = 3^x

Exponential functions are models for population, inflation, carbon dating and radioactive decay.  The sharp rise in EX2 graph indicates exponential growth.

The base b is any real number, not 1.  It can be rational (can be put into a fraction (ratio), thus it is either a terminating or repeating decimal).  Or irrational, not terminating or repeating decimal.  

Examples of irrational numbers: , ,  ~ (pi) . .

There is another irrational number that occurs in nature important enough to have a special name, 

        e = 2.71828182. . . .

Example3: 

Use calculator to find

        y = e^x  natural exponential function.

When

        x = 1;              y =  e   = 2.718 . .

        x = 2               y =  e² = 7.389 . .

        x = -1              y = e^-1 = 0.3678 . .

        x = -2              y = e^-2   = 0.1353

        x = 3               y = e³   = 20.085

Logarithms are inverses of the exponential function.

What is the exponent of 10 to get 100? to get 1000?  To get 346?

  10² = 100       so     x = 2

10³ = 1000     so       x = 3

10^x = 346                  x = ?

We can’t be precise, but we can state 

        2 < x < 3.

Definition of logarithm:  log_b u = v    means            b^v = u

So we can rewrite 

        10^x = 346      as      log₁₀346 = x

Since our number system is base 10, log of base 10 is called the common log and we do not have to write the subscript.

10^x = 346   can be rewritten as        log346 = x

We can use our calculators to solve.

Example 3:  

3 raised to what number gives us 9?          

        3^v = 9    

Or     

        log₃9 = v;        v = 2

Find u:             a) u = log₂8

                        b) log₅u = -2

                        c) log_u9 = 2

Sketch 

        y = log x         

which means         

        10^y = x

Logarithms are used in the measurement of earthquakes (the Richter Scale) and sound (decibel scale).  It measures small increments of x with large differences in y for 0 < x < 1 but for x > 1 the y values are compressed.

The natural log function has base e, the natural base.

        y = log_e x  

  is written 

        y = ln x  

can be rewritten         

        x =  = e^y

Note: the independent variable in a log function is greater than 0!

Example 4:  

Use calculators to Find: 

a) ln .34                 b)  log 2.3                   c) ln (-1.5)

d)  4.9 = log x        e)  ln x = - 2.1

Back to Exponenials and Logarithms Main Page

Back to the Survey of Math Ideas Home Page

Back to the Math Department Home Page

e-mail Questions and Suggestions