Converting Logs to Exponential Form

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) = ax + 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) = 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).
Exponential Graph


Logarithmic

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.

(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 = 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.

Helpful Conversion Chart

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


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.

1.  For any positive numbers x and y, loga (x * y) = loga x + loga y when a <> 1.  Example:


   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

5.  loga 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: 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)

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