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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. Scroll down or click on any of the links
below to gain a better understanding of exponential and logarithmic
functions.
Exponential functions Logarithmic functions Exponential and logarithmic equations Quiz on Exponential and Logarithmic 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 are functions where f(x) = ax + B
where a is any real constant and B is any expression.
For example, f(x) = e-x - 1 is an
exponential function.
1. Problem: Graph f(x) = 2x.
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 (shown below).
Logarithmic functions are the inverse of exponential functions.
For example, the inverse of y = ax is y = logax,
which is the same as x = ay.
1. Problem: Convert to logarithmic form:
8 = 2x
Solution: Remember that the logarithm
is the exponent.
x = log2 8
2. Problem: Convert to exponential form:
y = log3 5
Solution: Remember that the logarithm
is the exponent.
3y = 5
The figure below is a little chart that always helped us remember
how to convert from exponential to logarithmic form
and from logarithmic to exponential form.
3. Problem: Solve log2 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.
4. Problem: Simplify: log2 x + log2 6.
Solution: log2 (x * 6)
2. For any positive numbers x and p, loga xp = p * loga x. Example:
5. Simplify: logb 9-x
Solution: -x * logb 9
3. For any positive numbers x and y, loga (x/y) = loga x - loga y. Example:
6. Problem: Express as a single logarithm:
loga x - 5loga y
Solution: loga x - loga y5
(Using the 2nd rule.)
Use the third rule in reverse.
loga (x/y5)
4. loga a = 1
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: 3x = 8.
Solution: Take the logarithm of both
sides.
log 3x = 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:
2. Problem: Solve log3 (5x + 7) = 2 for x.
Solution: Write an equivalent exponential expression.
5x + 7 = 32
5x + 7 = 9
Solve for x.
5x = 2
x = (2/5)
Take the Quiz on exponential and logarithmic functions. (Very useful to review or to see if you've really got this topic down.) Do it! |
