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3.5  Graphical checks for examples

Example 3a.

We have a problem here because our graphing calculator has logs only to the base 10 and base e.  Since this equation has a log to the base 5, we need to somehow convert the equation to one containing only base 10 or e.  Watch this:

Given:   

Write in exponential form:                                                     (5.1)

(Though the answer is obviously correct here, we continue with the graphing method to make a point.  In practice, the algebraic check is more efficient in this case. )

Take the log of both sides:                                       

 (This is a crucial new step!  Since we are taking the log of both sides, this new equation is equivalent to equation 5.1)

So now we graph  and look for the point of intersection:

   We see that the graphs intersect when .

Example 3b.    

Again, we need to convert to a logarithm with a base of 10.

Write in exponential form:                                                          (5.2)

Take the log of both sides:                                          

(This again is a crucial  step!  Since we are taking the log of both sides, this new equation is equivalent to equation 5.2)

  So now we graph  and look for the point of intersection:

We see that the graphs intersect when .

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 Example 3c.  

        Again, we need to convert to a logarithm with a base of 10.

        Write in exponential form:                                                           (5.3)

        Take the log of both sides:                                       

        (This again is a crucial step !  Since we are taking the log of both sides, this new equation is equivalent to equation 5.3)

     So now we graph    and look for the point(s) of    

     intersection:

We see that the graphs intersect when .

AN interesting exercise would be to convert all of the above logs to natural logs (base e) and see if the checks work out.  Try it!

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