Log Relationship

A verbal description of the relationship

The dependent variable (Y) is related to the exponent of the independent variable (X). The function log(X) returns the exponent to the base 10. Thus Y = log(X) also means that 10^Y = X.

The equation and the meaning of x, y, and other parameters

Y = m*log(X)

X should be dimensionless in this equation. Log functions are often used with ratios, see below.

One or more examples

Imagine that road services have to increase by 5 times for each 10 fold increase in population. How much have the road facilities expanded from 1900 to the present. In this example X equals the ratio of increase in a population current_population/1900_population. The log of this ratio tells how many times the population has increased by 10 x. For example if the population was now 200,000 and it was 10,000 in 1900 then this has increase by 200,000/10,000 or 20 times. The log of 20 is about 1.3 and thus the road services only had to increase by 5*1.3 or 6.5 times.

The "m" factor is in terms of road services, the same units as Y.

A graph

Algebraic rules that apply to the use of this equation

The rules for manipulating logs and exponents are the same.

Remember that Y = 10^X is the same relationship as log(Y) = X. For example, 100 = 10^2, and 2= log(100).

The log of a product is the same as adding the logs. For example, the log(10*100) = log(10) + log(100) = 1 + 2 = 3

Characteristic values

for the equation Y = m*log(X)

at X can't equal 0

at X = 1, Y = 0

at X = 10, Y = m

increasing X by 10 times with only increase Y by 1*m