Pre-Calc             Hour ____                                            Name ______________

Application of Logarithms and Exponentials to Food Technology

A study is done in which the number of bacteria in a liquid culture is monitored over a period of several hours. Ten observations were made:

Two questions need to be answered:

  1. Is this data consistent with an exponential growth model: N= Aert, where

·        A = the number of bacteria present initially (at t = 0)

·        r = a growth rate constant

·        N = the number of bacteria present at a given time t,

·        e is the fundamental mathematical constant forming the base for natural logarithms.

  1. If the answer to question (1) is "yes", then what are the values of N, r, and t2 (the so-called doubling time) for this bacterial system?

The experimental measurements listed above indicate that the number of bacteria are growing with time. Plot the observed values of N (vertically) against the elapsed time, t, in hours, on your graphing calculator and carefully sketch the graph on the grid provided. Label all axes and scales.

 

 

 

 

 

 

 

 


 

1) Take two points from the data and find the exponential model for the data by hand. Show work below and indicate your equation.

y = ____________________________

 

 

 

2) Find the ExpReg using the graphing calculator and indicate your calculator's equation.

y = ____________________________________

3) Put both equations in your calculator and graph them with the data.

As you can see, we have two smooth curves through the experimental points, and the points all seem to lie relatively close to this curve. Further, these curves look quite a bit like the graph of an exponential function, but from what we see, it might just as well be part of a parabola or of one of a number of other curves that result when various algebraic forms are graphed. This graph suggests that the answer to (1) may be "yes", but we would usually want stronger evidence than this to draw a definite conclusion. The trouble here is that it is difficult to distinguish precisely and reliably between the curves are the graphs of quite different mathematical functions.

 There is, of course, one curve that is easily recognized by everyone – the straight line! In this situation, the logarithmic function can be used to rewrite the exponential growth formula into a form that will allow us to create a plot that does give a straight line when data is the result of an exponential growth (or decay) process. Start with the exponential growth formula:

                  N = A ert

Using the properties of logarithms, take the natural logs of both sides to change this exponential function into a linear equation.  Show your steps below.

 

 

 

You should have used two of the three basic properties of logarithms to manipulate the right-hand side of the equation, namely that

log (AB) = log A + log B        and               log (Am) = m log A

If the original exponential growth formula is correct and we plot ln N along the vertical axis of the graph (rather than N itself), and t along the horizontal axis, then the equation of the plot we are preparing will have the generic form

vertical quantity = constant x horizontal quantity + another constant

But this is just the equation of a straight line. Go to L3 and take the natural log of all the values of N.  Now plot ln N vs t on your graphing calculator and carefully sketch the graph that results using the data given above.

 

 

 

 

 

 

 

 


The graph indicates that the points are scattered about a straight line, the scatter due to small random errors in measurement. Since the plot of ln N vs t gives a straight line, we have confirmed that

ln N = rt + ln A                           (equation *)

is a valid formula for this data, and therefore, so is its equivalent, N = A ert, valid for this data. This ends our answer to the first question posed.

 It is now easy to answer the second question posed. Since we have concluded that (*) correctly describes the straight line graph in the plot, we notice that the constant r is just the slope of this line. To calculate the slope, we first read the coordinates of two points on the line. Working with some care, take two points from the graph. (for this sort of work, it’s important to pick your points as widely spaced as possible). Indicate them below.

r = slope = actual rise

                  actual run

Notice that the ‘actual rise’ has to be the actual vertical distance between the two points, and because the vertical scale is ruled off in proportion to ln N, this vertical distance will be the difference between the natural logarithms of the N values for the two points.  Using the slope formula above, find the slope of the line and the y-intercept.

 

Using your calculator, find the LinReg of the points and indicate the calculators equation.

 

How do the two equations compare?

 

Finally, compute the doubling time, t2, for this system, the time required for the number of bacteria to double.

 

 

Write a summary of the findings for this activity on the back of this paper.<Small> </Small>