Equation $Q=kH^n$

Hello i have a maths problem i am trying to solve. I have nearly completed it. I am just stuck on the last bit, and looking for some help.

My Question:

Two quantities Q and H are believed to be related by the equation $$Q= kH^n$$.

The values obtained for Q and H shown in the table below were obtained during an experiment.

Q

0.16

0.20

0.27

0.34

0.40

0.47

0.55

H

1.14

1.78

3.24

5.14

7.11

9.82

13.44

Plotting the values of Q and H using graph paper or computer graphing software show the relationship between Q and H and determine;

a. the gradient of the curve from your graph,

b. the law connecting Q and H, expressing the law that you have determined in the form of an equation.

So far i have drawn the graph and plotted all the points. I got the points like this:

$$Q = KH^n$$

$$Y = ax^n$$

$$Log (y) = Log (ax)$$

$$Log (y) = Log (a) + Log(xn)$$

$$Log (y) = Log (a) + nLog (x)$$

$$Log (y) = Log (a) + nLog (x)$$

[ y ] [ c ] [ x ]

1: Log (0.16) = -0.796

Log (1.14) = 0.0569

2: Log (0.20) = -0.699

Log (1.78) = 0.250

ETC..

I am now stuck on finding the graidient, and connection law. I know n is the graidient, just not how to find it.

Is the connecting law Logarithms?

Any help much appreciated, been stuck on this for a long time.

• gradient, not graidient. – Yves Daoust Mar 25 '19 at 8:58

On a bilogarithmic plot, you expect a straight line,

$$\log Q=n\log H+\log k.$$ This is indeed what you observe.

As the alignment is excellent, you can simply use the two extreme points.

Then

$$n=\frac{\log Q_7-\log Q_1}{\log H_7-\log H_1}$$ and $$\log k$$ follows.

• So i did $n=\frac{\ 0.259 - 0.796}{\ 1.128 - 0.0569}$. $n = 0.501$ .So the Gradient is 0.501 – Kyle Anderson Mar 25 '19 at 9:28
• Or is the Gradient $-0.537$ in $Q$. $1.0711$ in $H$? – Kyle Anderson Mar 25 '19 at 9:33
• @KyleAnderson: no idea how you obtain this. – Yves Daoust Mar 25 '19 at 9:53
• I obtained it by places the values in the equation you gave me. What have i done wrong? – Kyle Anderson Mar 25 '19 at 9:55
• @KyleAnderson Can't make sense of "Or is the Gradient −0.537 in Q 1.0711 in H?" Take $n=0.5$. – Yves Daoust Mar 25 '19 at 10:00