How to calculate gradient logarithmic scales

Updated April 17, 2017

The gradient of a straight line on a logarithmic plot can be calculated in a couple of different ways depending on whether the data in the plot was derived getting points from an equation or from a table of values. Equations in the form y=a*x^b are straight lines on logarithmic plots, which means the gradient can be obtained by applying the same methods as those used for straight lines. Logarithm is another word for index or power. For example, 2 is the logarithm of y=10^2. This equation in logarithmic form is 2=log-to-base-10(y). The number 10 is called the base. Logarithmic plots vary with respect to base 10.

Graph plotted based on an equation of the form y = a*x^b

Take the logarithm of both sides of your equation. In order to produce a straight line on a logarithmic plot, your equation is of the form y=a_x^b. Taking the logarithm of both sides gives log(y)=log(a_x^b).

Expand the equation. Using the property that log(f_g) = log(f) + log(g), this equation becomes log(y) = log(a) + log(x^b). This can be expanded further using the relation log(x^b)= b_log(x). Now the equation is log(y) = log(a) + b*log(x).

Calculate the slope from the equation log(y) = log(a) + b*log(x). This equation is of a similar form to the equation of a straight line y= bx + a, where b is the slope and a is the y intercept. The slope of the line is the constant b, which corresponds to the value of the exponent in the equation that you plotted.

Graph plotted with data points from a table

Pick any two points on the line and write down the coordinates.

Label the coordinates. Call of the first point x1 and y1 and the second x2 and y2, where x represents the horizontal axis and y the vertical.

Calculate the gradient. The gradient is (y2-y1)/(x2-x1).

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