Algebra II: Exponential and Logarithmic Functions

On this page we hope to clear up problems you might have with exponential and/or logarithmic functions. Logarithms are used all the time in real life, for example, the Richter Scale, and are very useful for measuring things that grow or diminish exponentially. Read on or follow any of the links below to gain a better understanding of exponential and logarithmic functions.

Inverse functions
Exponential functions
Logarithmic functions
Exponential and logarithmic equations
Quiz on exponential and logarithmic functions

Inverse Functions

To find the inverse of a function (the inverse of a function is the same as reflecting a function across the line y = x), interchange x and y and then solve for y. The inverse of f(x) is denoted by f^(-1)(x).

Example:

1.  Problem: Find f^(-1)(x) of 3x + 1.
   Solution: The equation is y = 3x + 1.  Interchange x and y.

             x = 3y + 1

             Solve for y.

             x - 1 = 3y
             (x - 1)/3 = y

             f^(-1)(x) = (x - 1)/3

Exponential Functions

Exponential functions are functions where f(x) = a^x + B where a is any real constant and B is any expression. For example, f(x) = e^(-x) - 1 is an exponential function.

To graph exponential functions, remember that unless they are transformed, the graph will always pass through (0, 1) and will approach, but not touch or cross the x-axis.

Example:

1.  Problem: Graph f(x) = 2^x.
   Solution:  Plug in numbers for x and find values for y, as
             we have done with the table below.

             _____________________
             | x | 0 | 1 | 2 | 3 |
             ---------------------
             | y | 1 | 2 | 4 | 8 |
             ---------------------

             Now plot the points and draw the graph (see accompanying figure).

Logarithms

Logarithmic functions are the inverse of exponential functions. For example, the inverse of y = a^x is y = log(SUB a) x, (The "symbol" SUB means that anything after it would be subscripted if text only browsers allowed for subscripted characters.) which is the same as x = a^y.

(Logarithms written without a base are understood to be base 10.)

This definition is explained by knowing how to convert exponential equations to logarithmic form, and logarithmic equations to exponential form.

Example:

1.  Problem: Convert to logarithmic form: 8 = 2^x.
   Solution: Remember that the logarihtm is the exponent.

             x = log(SUB 2) 8

2.  Problem: Convert to exponential form: y = log(SUB 3) 5
   Solution: Remember that the logarithm is the exponent.

             3^y = 5

This figure is a little chart that always helped us remember how to convert from exponential to logarithmic form and from logarithmic to exponential form.

Sometimes you can solve equations containing logarithms by changing everything in logarithmic form to exponential form.

Example:

1.  Problem: Solve log(SUB 2) x = -3.
   Solution: Convert the logarithm to exponential form.

             2^(-3) = x
             x = (1/8)

There are five special rules that you ought to always have in mind when working with logarithms. They will help you in such tasks as simplifying expressions containing logarithms and solving equations containing logarithms. They are outlined below.

1. For any positive numbers x and y, log(SUB a) (x * y) = log(SUB a) x + log(SUB a) y when a <> 1

Example:

1.  Problem: Simplify: log(SUB 2) x + log(SUB 2) 6.
   Solution: log(SUB 2) (x * 6)

2. For any positive numbers x and p, log(SUB a) x^p = p * log(SUB a) x.

Example:

1. Simplify: log(SUB b) 9^(-x)
   Solution: -x * log(SUB b) 9

3. For any positive numbers x and y, log(SUB a) (x/y) = log(SUB a) x - log(SUB a) y.

Example:

1.  Problem: Express as a single logarithm:
             log(SUB a) x - 5log(SUB a) y
   Solution: log(SUB a) x - log(SUB a) y^5  (Using the 2nd rule.)

             Use the third rule in reverse.

             log(SUB a) (x/(y^5))

4. log(SUB a) a = 1

5. log(SUB a) 1 = 0

Exponential and Logarithmic Equations

An equation with variables in its exponents is called an exponential equation. To solve these, take logarithms of both sides and use theorems 1 - 5 listed in the section above to simplify and then solve for x.

Example:

1.  Problem: Solve for x: 3^x = 8.
   Solution: Take the logarithm of both sides.

             log 3^x = log 8

             Use theorem 2 to simplify the equation.

             x * log 3 = log 8

             Solve for x by dividing each side by log 3.

             x = (log 8/log 3)

             A decimal approximation may be found if desired.  x = 1.8929.

To solve logarithmic equations, you convert them to exponential form and solve for x.

Example:

1.  Problem: Solve log(SUB 3) (5x + 7) = 2 for x.
   Solution: Write an equivalent exponential expression.

             5x + 7 = 3^2
             5x + 7 = 9

             Solve for x.

             5x = 2
             x = (2/5)

Take the quiz on exponential and logarithmic functions. The quiz is very useful for either review or to see if you've really got the topic down.

Back
Site Map
Home
Next

Math for Morons Like Us -- Algebra II: Exponential and Logarithmic Functions
/20991/textonly/alg2/log.html
© 1998 ThinkQuest Team 20991