1. The Power Function Studies of stream length and drainage basins determined an empirical relationship : L = 1.4 A 0.6 where L is stream length and A is drainage basin area. (Hack, 1957)

2. The Power Function In our lab exercise on sinking forams, you derived the equation for Stokes settling velocity, V stokes = (  g/18  d 2 Stokes derived this equation from consideration of the driving forces and resisting forces for sinking foraminifera.

3. The Power Function Both the empirically defined Hack equation and the analytically derived Law for Stokes velocity are examples of power functions. A power function is written in general form by y = ax b.

4. The Power Function In the case of Hack's Law, y = L x = A b = 0.6 (a = 1.4) L = 1.4 A 0.6 y = ax b

5. The Power Function In the case of Stokes velocity, y = V stokes x = d b = 2 (a =  g/18  ) y = ax b V stokes = (  g/18  d 2

6. The Power Function y = ax b What is interesting about this equation is what happens when you apply logarithms. How would you do it ? log y = log a + b*log x Does this equation remind you of anything ?

7. Logarithm of a Power Function log y = log a + b*log x This may be similar to the equation for a straight line where y = b + m*x b = 3 m = 1 y = 3 + 1*x y = 3 + x What kind of scale would we need to plot the logarithmic equation to simulate a linear equation ? (at the x intercept when x = 0, y = 3)

8. Logarithm of a Power Function log y = log a + b*log x If a and x are a set of measurements and y is a column of results and we take the log of each of these numbers Then plot these log values on normal graph paper... We see a straight line. With b as the slope. The x intercept (log x = 0) occurs at log a x y

9. Logarithm of a Power Function log y = log a + b*log x If we plot x against y on log-log paper, We also see a straight line Again, b is the slope The line crosses x = 1 Where y = a x y

10. Power Functions in Geology Log - log plots are common in geology As a result, power functions often arise in geology C = C o F (D-1) As crystals settle out of a magma element concentrations, C, in the remaining liquid change according to this equation. Where Co is initial concentration, F is the fraction of liquid remaining, and D is the distribution coefficient. Linear plot of C = C o F (D-1)

11. Power Functions in Geology Log - log plots are common in geology As a result, power functions often arise in geology C = C o F (D-1) log-log plot log C = log C o + (D-1) log F

12. Power Functions in Geology Stream length (y) and drainage-basin area (x) are measured and listed in the table above. The logs of each measurement are listed in column 4 and 5 If we plot columns 4 and 5 and try to “fit” a line to the data Constant = 0.148761, and slope is 0.53687

13. Power Functions in Geology Constant = 0.148761, and slope is 0.53687 How can we write this in a linear style equation with logs ? log y = 0.148761 + 0.53687 log x

14. Power Functions in Geology log y = 0.148761 + 0.53687 log x Plot columns x and y (squares) Test theory, but plotting the line for the log eqn above. Pretty good fit!

15. Power Functions in Geology log y = 0.148761 + 0.53687 log x Remember that if we take the “antilog” of both sides We get y = 10 0.148761 x 0.53687 Simplifying, y = 1.41 x 0.54

16. Power Functions on a Linear Scale y = 1.41 x 0.54 Data in a power function plotted on a linear-linear scale The curve continues to increase But it increases at an ever decreasing slope

17. Power Functions on a Linear Scale y = 1.41 x 0.54 To understand the “slopes” of a function, take it's derivative dy = 0.76 x -0.46 dx The exponent, b is < 1 (negative) This says the slope will decrease, as x progresses rise run =

18. Power Functions on a Linear Scale Taking the derivative in general dy = (a) x b-1 dx If the exponent, b is > 1 (positive) Then the slope will increase, as x progresses What if b = 1 ? Then what ?

19. Power Functions on a Linear Scale y = 1.41 x 0.54 To Summarize: For y = ax b-1 Plots will be convex- upward if b < 1 (negative exp) Plots will be convex -downward if b > 1 (positive exp) Plots will be a straight line if b = 1.

20. Power Functions and Exponential Functions It is easy to confuse power fns with exponential fns We've already looked at exponential functions But we have not studied power functions until today. y = x b y = b x Exponential functions produce a straight line when plotted on a linear-log scale. Where as power functions produce a straight line when plotted on a log-log scale

21. Power Functions and Exponential Functions y = x b In a power function, for every increase in x by some factor y increases by some other factor In an exponential function, for every increase in x by some factor y may increase by an order of magnitude (assuming b is a whole number) This is where the concept of a half-life comes from. y = b x

22. Studies of stream length and drainage basins determined an empirical relationship : L = 1.4 A 0.54 where L is stream length and A is drainage basin area. (Hack, 1957) Back to Drainage Basins and Hack's Law The exponential “b” value here has been debated.

23. L = 1.4 A 0.6 Back to Drainage Basins and Hack's Law Some say that if b > 0.5 Then the length/area relationship implies that large basins are more elongated.

24. L = 1.4 A 0.54 Shape of Drainage Basins Understanding length/area ratio If A = wL Then, L = 1.4 (wL) 0.54 Simplifying.... w/L = 0.53L -0.15 w L Notice that the exponent is negative. How will w/L change as you go downstream (increasing L) ?

25. Shape of Drainage Basins Put L on one side: w/L = 0.53L -0.15 w = 0.53 L 0.85 Will a plot of L versus w be convex up or down ? w L