Question

Two variable quantities x and y are related by the equation

where a and k are constants.

Four pairs of values of x and y are measured experimentally. The result of plotting ln y against x is  shown in the diagram. Use the diagram to estimate the values of a and k.

Solution

We are given the equation;

It is evident that it is an exponential equation but it is plotted in logarithmic form  and,  therefore, we can write it in logarithmic form as follows.

Taking natural logarithm of both sides;

We can rearrange the equation as;

Slope-Intercept form of the equation of the line;

Where  is the slope of the line.

Comparing the obtained equation with slope-Intercept form of the equation of the line;

We have expressions for both slope and intercept of the given line.

We can also find both slope and intercept from the given graph and equate with these expressions.  First we deal with intercept of the line.

From the graph we can see that y-intercept of the line;

Taking inverse logarithm of both sides;

 and  are inverse functions. The composite function  is an identity function, with  domain the positive real numbers. Therefore;

Therefore;

Next we deal with slope of the line.

Expression for slope of a line joining points  and ;

We can find coordinates of two points on the line from graph as  and . Therefore;

Hence;

Taking inverse logarithm of both sides;

 and  are inverse functions. The composite function  is an identity function, with  domain the positive real numbers. Therefore; 

Therefore;