Natural Logarithmic Function and Solving Logarithmic Functions and Exponential Functions

Here is a graph (courtesy of wikipedia.org) that illustrates the graph of ln(x) (the green line).  As can be seen from the graph, as you move to the right from the origin, the graph above the y=0 line starts to increase and move upward.  At the same time, moving from right to left, the graph approaches a vertical asymptote at x=0.

Solving exponential equations -

 For this part, we're going to look at the equation  .

The first thing to do is to tak the Ln of both sides.

That leaves us with the following equation.

The Ln(e) cancels out, and we are left with -

which can easily be solved on a calculator.

Solving a logarithmic equation such as -

Move the 4 to the other side of the equation.

Make the logarithmic function into an exponential function by making it 3^6

Divide each side of the equation by seven.

And there you have it.

The Graph of f(x)=a^x

The graph of f(x)=a^x is defined as being an exponential function. That just means that the function contains a variable expressed as an exponent.



Graph courtesy of http://www.bced.gov.bc.ca/irp/mathk7/icons/f26.gif.


As can be seen from the graph, there is a horizontal asymptote at y = 0, and a vertical asymptote at x = 3. It passes through the points (2, 3.2) and (-1, .75).

There are some phase shift properties to be aware of.

a) f(x)= (a^x)- 1 -> The graph looks just the same, excpet that the graph and horizontal asymptote are shifted down one unit.

b) f(x)= a^(x-1) -> The graph is the same as a^x but it is shifted one unit to the right.

c) f(x)= -a^x -> The graph is the exact same (not moved up or down or left to right) but is flipped directly across the x-axis.

Vertical and Horizontal Asymptotes

A horizontal asymptote is an imaginary line drawn across a coordinate plane that acts as a limit to a function.  In other words, if there were a horizontal asymptote at y = 3, as x moved forward the graph would continue to approach y = 3 but would never actually reach 3.  Here is an example from www.wikipedia.org that illustrates my point.  

Notice how the lines of the graph approach the horizontal dotted lines (the asymptotes) but never reach them.

A vertical asymptote is very similar, but instead of being horizontal lines that limit the graph, they are vertical lines that help to give shape to the graphical representation of the function.  Here is another picture (also from www.wikipedia.org) to better explain what I mean.  

In this case, the y-axis acts as the asymptote.

Finding the asymptotes is relatively simple.

For a horizontal asymptote, we write the equation in standard form such as  and we ignore everything except the numbers with the highest powers (the x^2 and the 4x^2) and divide them as normal.  The x^2 cancels out, and we are left with our horizontal asymptote, 1/4.

For vertical asymptotes, we again write the equation in standard form  and this time just find the zeroes of the denominator.  Just set the denominator to zero and solve the resulting equation.  

4x^2 - 9 = 0

4x^2 = 9

x^2 = 9/4

x = plus or minus 3/2

So in this case, we have a vertical asymptote at x = 3/2, another at - 3/2, and a horizontal asymptote at y = 1/4.

That's how it's done.