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

Definitions f(x) = a x, exponential function with base a f(x) = log a x, logarithm of x base a f(x) = ax b, is an allometric function

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

Population growth is a classic example Algae : cell division Geometric growth … t =

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

Special case, a = 1

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

f(x) = a x f(x) = log a x

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

Laws of logarithms log a (xy) = log a x + log a y log a (x/y) = log a x - log a y log a x k = k·log a x log a a = 1 log a 1 = 0 Example 15.7 :

Example 15.8 : Radioactive decay 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?

To compute log a x if your calculator doesn’t have log a Example: or use

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) =  e mx

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

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

Example 17.9 : Wound healing rate Regular plot Semilog plot

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)

Estimate slope and intercept using least squares Y = b+mx m = b = 4.69 b = ln  ->  = e b = e 4.69 = m = ln  ->  = e m = e = A = (0.953) t ln(A)= t A=108.85(0.953) t

f(x)=bx a 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.

a > 1 a = 1 a < 1 a < 0

f(x) = ba x f(x) = bx a

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 ft 2 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 = bx a = bx = b(1) = b y=20x 0.74 y=20(3.3) 0.74 =48.4 ft 2

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

log-log plot Semilog plot Regular plot

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.

Y=bx a ln(y)=ln(bx a ) ln(y) = ln(b) + ln(x a ) 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.