We discussed in the previous post that a straight-line graph is very helpful in interpreting experimental data. However, sometimes experiments give us data that can only be represented as curves. These curves are not very helpful as they do not give us the exact relationship between our two variables x and y. In this post, we shall discuss the procedure for getting relevant information from these curves.
Different Relationships between ‘x’ , and ‘y’
x and y can be related in the following two ways –
1) FIRST DEGREE EQUATION–
The equation that defines a linear dependence between variables, is called the equation of the first degree. It is called so as all the variables in this equation are raised to the power one only.
y = mx+c.
This is a linear equation that gives us a straight line.
2) SECOND/ HIGHER DEGREE EQUATION-
Many times a physical situation can only be defined by a second/higher degree equation. This is because the variables change exponentially with respect to each other. A higher-degree equation can be written as follows –
y = Axn
There is an exponential relationship between the two variables x and y. This equation gives us a curve.
Point to remember –
y=mx+c ⇒ Straight Line ⇒ Linear relationship
y=Axn ⇒ Curve ⇒ Exponential relationship
In the exponential equation, y= Axn,
If n=2, the equation is a quadratic equation i.e y=Ax2
If n=4, the equation is a quartic equation i.e. y=Ax4.
Both these equations give us 2 different curves on the graph, which look similar, as shown in the following figure-
So, if we plot experimental data on a graph paper and get a curve, we cannot determine the exact relationship between x and y (it could be y=Ax2 or y=Ax3 or y=Ax4, etc). Thus, curves are not very helpful in the interpretation of data as they tell us nothing accurate. However, straight lines indicate the exact relation between the two variables under study. So, is there a way to convert the curves to a straight line? The answer is YES! We can do that by taking logarithms on both sides of the exponential equation!
Log-log graphs.
Before proceeding with the topic, we need to learn some basic rules of logarithms.
1) THE PRODUCT RULE– The logarithm of the product of two numbers x and y is equal to the sum of the logarithms of x and y.
log xy = log x + log y.
2) THE POWER RULE– The logarithm of x raised to the power n is equal to n multiplied by the logarithm of x.
log xn = n log x
3) THE QUOTIENT RULE – The logarithm of the quotient of two numbers x and y is equal to the difference of logarithms of x and y.
log (x/y) = log x- log y
Now let us get back to our equation-
y = Axn
Taking logs on both sides, we get,
log y = log xn +log A
∴ log y = n log x + log A
log y = n log x + log A, is like the linear equation, y = mx + c. ‘log A’ is the y-intercept and a constant. The graph will be as follows –
If A=0 (y-intercept) when the line passes through the origin.In that case, log y = n log x. This is a linear equation of type y = mx!
If we plot a graph of log y vs log x, when A=0, we would get a straight line passing through the origin!
Note ⇒ In the above graph, Slope (m) = ‘n’ the exponential factor.
The slope (m) of this straight line can be determined by choosing any two points on the line and plugging in values in the equation, m = (y2-y1 / x2-x1). The slope is nothing but the exponential factor ’n’.
Thus, if the slope is 2, we know y=x2; if the slope is 3, y=x3
A logarithmic function has an exponential function as its inverse.
y = log 10 x implies y =10x and
ln e x implies , y = ex.
Thus for exponential functions, we take the logarithm on both sides and convert the equation into a linear one. We further find the slope of the line and determine the value of the exponential factor n. After taking the log, if we do not get a straight line, the two variables are not exponentially related.
Point to remember → We take the logarithm of an exponential function to get a straight line on the graph.
We shall study these kinds of log-log graphs later in Chemical Kinetics and Nuclear Chemistry in future posts. Till then,
Be a perpetual student of life and keep learning…
Good Day!
Madoverchemistry 