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Morrison on Metrics: Understanding business using mathematical relationships and functions

Linear and exponential functions yield different results.

This column is about a way to understand and describe certain mathematical relationships: exponential functions expressible with logarithms. Think of a trend line on a scattergram as a visual math function. The equation for it that Excel produces is a function – a mathematical connection that links one number to a second number in an orderly and specific manner. There can be many functional relationships, including linear and exponential, the latter showing the familiar hockey stick of increases seen with population growth and power laws.

With a linear function, the numbers change in lockstep: one more year of practice and $10,000 more in bonus. One more law firm retained; one more matter opened in a database.

By contrast, with an exponential function, one number rises faster than stepwise: $1 million more in fees for a law firm partner above the first million and compensation doesn’t double, it triples. The smaller a company’s revenue, the more it spends on legal costs as a percentage of that revenue. The more lawsuits are filed against the same company regarding a product, the higher the defense and settlement costs soar. The more offices a law department opens, the greater the overhead expenses. If all of those increases accelerate, an exponential function expresses it, not a linear function. The rate of change increases for an exponential function but remains the same for a linear function.

It’s important to understand that when you convert one of the sets of numbers to a logarithm, where there is an orderly exponential function, the trend line becomes straight instead of curving in the familiar upward sweep like the letter J.

Exponential functions involve logarithms. Sometimes called "logs,” they express one number in terms of a "base" number that is “raised” to some power. The power is called the exponent, which is the logarithm. If you understand logarithms, you will make more sense out of some kinds of benchmark or numeric relationships.

Let’s catch up with some terminology. For instance, 10 multiplied by itself four times equals 10,000. As an equation 104 = 10,000, which also can be written as log1010,000 = 4. You read that as, "The logarithm of 10,000 with base 10 is 4." Four is the exponent to which 10 must be raised (multiplied by itself) to produce 10,000. "104 = 10,000" is called the exponential form while "log1010,000 = 4" is called the logarithmic form.

So-called common logs use 10 as the base so each increase of one in the exponent signifies a number 10 times larger. The Richter scale for earthquakes is a well-known example. Sometimes so-called natural logarithms, which use e as the base (e is named after the 18th century Swiss math genius, Leonhard Euler), serve better. The irrational number, e, has a decimal value of about 2.7183. Logarithmic functions with e are particularly good to show percentage increases.

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