Graphs of Logarithmic Functions - At A Glance

Imagine your world flipped upside down and backwards. That's what happened to the exponential function, and in this section we are exploring the inverse of an exponential function...drum roll please...The Graph of the Logarithmic Function. We're talking about the graphs of logarithmic functions, and how they have a vertical asymptote (compared to a horizontal one in exponential functions). We won't forget the good stuff like domain and range. We'll also give you a few tips of graphing these on technology.

Since we have already decided that logarithms were the inverses of the power function, we can see that when we graph these, they are symmetrical about the diagonal line y = x which makes a 45 degree angle with the origin. Huh?

Sample Problem

Graph y = 10^x, y = log₁₀x, and y = x for values greater than 0.

Okay, here's a play by play of the above graph:

• The red line is y = 10^x.
   
• The blue line is y = log₁₀x.
   
• The green line is y = x.
   
• The red line and blue line don't like each other.

Every function's inverse is a reflection over the diagonal line y = x. To think of this in the real world, picture yourself leaning on a counter at a 45 degree angle and looking into the mirror. If you're lucky, you might be staring directly at your inverse. Or you may realize you need to clean the mirror.

Logarithmic graphs are just plain weird. They are different, for one thing, because they have a vertical asymptote rather than a horizontal asymptote.

Sample Problem

Graph y = log₂x for x > 0.

If you have a graphing calculator, you could plot this in your y-editor but you might have to convert the base 2 logarithm into base 10. Here's how you do that:

Instead of using:

y = log₂x

Use this:

In fact, any time you get the fancy to convert a logarithm into something you can use on the calculator, simply divide the log x function (which is naturally base 10) by the log of the base that you are in.

Sample Problem

Graph y = log₄x and y = log₄₀x and explain the differences:

We will graph  and  so each function will be in the same base (common logarithm which is base 10).

Here are some qualities of this graph above

• The red function is y = log₄₀x.
    
• The blue function is y = log₄x.
   
• The graph looks odd because we are showing you negative x values too, which happen to be imaginary values (we usually don't graph these) but we thought it would be fun to look at.
   
• The vertical asymptote is y = 0 for each function.
    
• Domain for each function: (0,∞), we didn't include the negative x values because the y values are not real numbers.
   
• Range for each function: All real numbers.
   
• Continuous? Yes, only if you are consider values of x > 0.
   
• Is the function increasing or decreasing? increasing if we just look at positive x values.
   
• Vertical asymptote: y = 0.
   
• Loves listening to Stephen Hawking and is fascinated with black holes.