Logarithmic Functions

What is a logarithm? A logarithmic equation is a “re-write” of an exponential equation. 

• Recall that the form of an exponential equation is .
• When we rewrite to logarithmic form, we are using the fact that exponential functions are one-to-one (pass the Horizontal Line Test), so they have an inverse function.
• We have the following equivalent statements (in symbols and verbally), each of which relate a specific value x to a specific value y when (x, y) is a point on the graph of .

Symbolic
Verbal	A base a is raised to a power x and gives an output of y	The logarithm of y to the base a is x.

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Example 1 - How do you convert exponential equations to logarithmic equations?

Example 2 - How do you convert logarithmic equations to exponential equations?

A handout called “Become Friends with Logarithms” prepared by a thoughtful colleague provides the kind of simple, repetitive practice that will clarify this concept of a logarithm. Click here to get a copy of the handout.

Example 3 - What does a logarithmic graph look like?

We have observed that the typical exponential function increases quite rapidly. Suppose we decided to switch the input and the output in the function. We would expect the reverse to happen. We would expect the "reversed" graph to increase very slowly for a base greater than one. This "reversed" graph (called the inverse graph) does indeed increase quite slowly. This is the typical logarithmic function.  Notice since the exponential and logarithmic functions are using the same base, they are inverse functions. This appears graphically as we see that the two functions are reflections across the line . Review the use and creation of inverse functions in the Inverse Functions lesson if this bothers you.

To graph the exponential and logarithmic functions on the same axes we have rewritten the exponential Then, we interchanged the use of x and y in the log statement.

In this busy little animation, we have graphed the family of curves related to.  We are allowing the base to change so that . Watch and you will how as the base is changed from a fraction less than one to greater than one, the logarithmic graph is changed from decreasing to increasing. Pay attention to the range of values. Logarithmic outputs can take on any real value. Also notice that in the domain, the logarithm can only work with positive numbers.

 

Example 4 - Modeling the Population of the US

Population of the US in Millions (Excluding Alaska and Hawaii) (from Statistical Abstracts, 1993)   Year Population Year Population 1790 3,929 1920 91,972 1810 7,240 1930 122,755 1830 12,866 1950 150,697 1870 38,818 1970 202,229 1890 62,948 1990 247,052

1. Shift the data so that 1790 represents year "zero." (Let x = 0 in 1790.)

2. Use your calculator to find an exponential model for the data. Do the following

   • Enter the data into your calculators statistical area as lists.
   • Plot the data as a scatter plot.
   • Use your calculator to find a fitting exponential function.
   • Round coefficients and the base to the nearest thousandth (3 decimal digits).

3. For the solutions slide the mouse over the black box in each column of the table.

See Data Lists	See Scatter plot	See Exponential Model