Precalculus I Class 03/13


During the preceding class we linearized temperature vs. clock time data by using a log(T) vs. t transformation.  

Here we will illustrate the general process we will use in this course to linearize data.  

The y vs. x table shown below is built using the y = 5 x^3 function.  Usually when we linearize data we don't know what the best function is (as was the case with the temperature data), but for this illustration we'll start with a known function.

A more accurate table is shown here:

x y log x log y
0 0
1 5 0 0.698970004
2 40 0.301029996 1.602059991
3 135 0.477121255 2.130333768
4 320 0.602059991 2.505149978

 Our graphs will be set up as illustrated below.

Graph of log(y) vs. x is not very linear, increasing at a decreasing rate:

The graph of log y vs. log x is pretty much linear, increasing at what appears to be a constant rate.

The straight line through these data points has an equation of form log(y) = m * log(x) + b, where m is slope and b is the vertical intercept.

Our results correlate with the power function we used to generate the data.  We haven't yet developed the tools to see precisely how these relationships exist.  However for future reference: 

For reasons that we'll see later, a power function y = A x^p will result in a log y vs. log x graph whose slope is p and whose y-intercept b has the property that 10^b = A.

We will see in upcoming classes why the following general rules for linearization hold:

 

We now investigate the idea of inverse functions. 

Our original function is y = x^3.  We obtain the formula for the inverse function as follows:

x = y^3.  We take the 1/3 power of both sides, obtaining

x^(1/3) = (y^3)^(1/3).  We use laws of exponents to simplify the right-hand side and get

x^(1/3) = y.  We switch the two sides to obtain

y = x^(1/3).

We sketch a graph of the function and the inverse function and note its significant characteristics:

The main characteristic of a graph of a function and its inverse is that the two functions are symmetric with respect to reflection about the line y = x.

An accurate plot of these functions is shown below:

'Zooming in' on the domain 0 <= x <= 3:

The log function y = log(x) is defined as the inverse of the y = 10^x function.

A table of y = 10^x and the 'inverted' table y = log(x) is shown below.

A rough graph of these functions is depicted below.  The graph in the lower-left-hand corner is an enlarged graph of these functions near the origin.

The figure below is a more accurate graph showing y = 10^x, y = log(x) and y = x.  Note the symmetry of these graphs with respect to reflection thru the line y = x.

A closeup of the same graphs near the origin shows the following important characteristics:

 

A closeup of the same graphs near the origin shows the following important characteristics:

A comparison between the graphs of y = 10^x and y = log(x), and the graphs of y = e^x and y = ln(x), reveals the following: