Exponentials, logarithms and rescaling of data

1. Exponentials, logarithms and rescaling of data Math 151 Based upon previous notes by Scott Duke-Sylvester as an adjunct to the Lecture notes posted on the course web page

2. Definitions • f(x) = ax, exponential function with base a • f(x) = logax, logarithm of x base a • f(x) = axb, is an allometric function

3. Motivation • Many biological phenomena are non-linear: • Population growth • Relationship between different parts/aspects of an organism (allometric relationships) • The number of species found in a given area (species-area relationships) • Radioactive decay and carbon dating • Many others • Exponentials, logs and allometric functions are useful in understanding these phenomena

4. Population growth is a classic example • Algae : cell division • Geometric growth 32 64 … 8 16 4 2 1 t = 0 1 2 3 4 5 6

6. Exponentialsf(x) = ax, a > 0 0 < a < 1 exponential decrease a > 1 exponential increase As x becomes very negative, f(x) gets close to zero As x becomes very positive, f(x) gets close to zero

7. Special case, a = 1

8. f(x) = ax is one-to-one. For every x value there is a unique value of f(x). • This implies that f(x) = ax has an inverse. • f-1(x) = logax, logarithm base a of x.

9. f(x) = ax f(x) = logax

10. logax is the power to which a must be raised to get x. • y = logax is equivalent to ay = x • f(f-1(x)) = alogax = x, for x > 0 • f-1(f(x)) = logaax = x, for all x. • There are two common forms of the log fn. • a = 10, log10x, commonly written a simply log x • a = e = 2.71828…, logex = ln x, natural log. • logax does not exist for x ≤ 0.

11. Laws of logarithms • loga(xy) = logax + logay • loga(x/y) = logax - logay • logaxk = k·logax • logaa = 1 • loga1 = 0 • Example 15.7 :

12. A radioactive material decays according to the law N(t)=5e-0.4t When does N = 1? For what value of t does N = 1? • Example 15.8 : Radioactive decay

13. To compute logax if your calculator doesn’t have loga or use Example:

14. general formula for a simple exponential function: • f(x) = x or • y = x • Then • ln(y) = ln x • ln(y) = ln  + ln (x ) • ln(y) = ln  + x ln  • Let b = ln , and m = ln , Y=ln (y) then this shows • Y = b + mx • which is the equation of a straight line. • This is an example of transforming (some) non-linear data so that the transformed data has a linear relationship. An exponential function gives a straight line when you plot the log of y against x (semi-log plot). Special exponential form : f(x) = emx

15. Consider the algae growth example again. How do you know when a relationship is exponential? Regular plot Semilog plot N= t ln(N) = mt+b

16. ln(N) = (0.693)t • Fit a line to the transformed data • Estimate the slope and intercept using the least squares method. • Y=mx+b • b~0, m = 0.693.. • Estimate  and . • b = ln ->  = eb = e0 = 1 • m = ln ->  = em = e0.693.. = 2.0 • N = 2t N=2t

17. Example 17.9 : Wound healing rate Regular plot Semilog plot

18. How do you make a semilog plot? • Use the semilog(x,y) command in Matlab • Take the log of one column of data and plot the transformed data (here log y) against the untransformed data (here x)

19. ln(A)=4.96-0.048t • Estimate slope and intercept using least squares • Y = b+mx • m = -0.048 • b = 4.69 • b = ln ->  = eb = e4.69 = 108.85 • m = ln ->  = em = e-0.048 = 0.953 • A = 108.85(0.953)t A=108.85(0.953)t

20. f(x)=bxa • Allometric relationships (also called power laws) • Describes many relationships between different aspects of a single organism: • Length and volume • Surface area and volume • Body weight and brain weight • Body weight and blood volume • Typically x > 0, since negative quantities don’t have biological meaning.

21. a > 1 a < 0 a = 1 a < 1

22. f(x) = bax f(x) = bxa

23. Example : It has been determined that for any elephant, surface area of the body can be expressed as an allometric function of trunk length. • For African elephants, a=0.74, and a particular elephant has a surface area of 20 ft2 and a trunk length of 1 ft. • What is the surface area of an elephant with a trunk length of 3.3 ft? • x = trunk length • y= surface area • y = bxa = bx0.74 • 20 = b(1)0.74 20 = b • y=20x0.74 • y=20(3.3)0.74=48.4 ft2

24. How do you know when your data has an allometric relationship? • Example 17.10

25. Regular plot log-log plot Semilog plot

26. How do you make a log-log plot? • Use the loglog(x,y) command in matlab • Take the log of both columns of data and plot the transformed columns.

27. Y=bxa • ln(y)=ln(bxa) • ln(y) = ln(b) + ln(xa) • ln(y) = ln(b) + a ln(x) • Let • Y = ln(y) • X=ln(x) • B=ln(b) • Then • Y=B + a X • Which is the equation for a straight line. So an allometric function gives a straight line on a loglog plot.