**Inverse**** Function**
**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 Form**
**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 (shown below).

**Logarithmic**
**Logarithmic functions are the inverse of exponential functions. For example, the inverse of ***y = a*^{x} is *y = log*_{a}x, 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. Examples:

**
1. Problem: Convert to logarithmic form:
***8 = 2*^{x}
Solution: Remember that the logarithm
is the exponent.
**x = log**_{2} 8
**
2. Problem: Convert to exponential form:
***y = log*_{3} 5
Solution: Remember that the logarithm
is the exponent.
**3**^{y} = 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. **
** **

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

**
3. Problem: Solve ***log*_{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*_{a} (x * y) = log_{a} x + log_{a} y when *a <> 1*. Example:

**
4. Problem: Simplify: ***log*_{2} x + log_{2} 6.
Solution: **log**_{2} (x * 6)

**2. For any positive numbers ***x* and *p*, *log*_{a} x^{p} = p * log_{a} x. Example:

**
5. Simplify: ***log*_{b} 9^{-x}
Solution: **-x * log**_{b} 9

**3. For any positive numbers ***x* and *y*, *log*_{a} (x/y) = log_{a} x - log_{a} y. Example:

**
6. Problem: Express as a single logarithm:
***log*_{a} x - 5log_{a} y
Solution: *log*_{a} x - log_{a} y^{5}
(Using the 2nd rule.)
Use the third rule in reverse.
**log**_{a} (x/y^{5})

**4. ***log*_{a} a = 1

5. *log*_{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:

**
2. Problem: Solve ***log*_{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)**