# Logarithmic growth vs exponential growth

## Main Question or Discussion Point

From the book Calculus made easy: "This process of growing proportionately, at every instant, to the magnitude at that instant, some people call a logarithmic rate of growing."

From Wikipedia: "Exponential growth is feasible when the growth rate of the value of a mathematical function is proportional to the function's current value, resulting in its growth with time being an exponential function" from another Wikipedia page: " In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y = C log (x)." "Logarithmic growth is the inverse of exponential growth and is very slow"

jbriggs444
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2019 Award
"Some people say" is not a very broad statement.

fresh_42
Mentor
I guess Wikipedia is right here, although both descriptions needed some explanations of what is actually meant.

Let's define the value of growth ##G(f;x_0)## of a function ##f## at an "instant" point ##x_0## as ##\left.\frac{d}{dx}\right|_{x=x_0}\,f(x)##.

Then for ##f(x) = \exp(\alpha x)## we get ##G(\exp;x_0)=\alpha \exp(\alpha x_0) = \alpha f(x_0) \sim f(x_0)## which shows the growth at this point is proportional to the function's value, if the function is the exponential function.

For the logarithm ##f(x)=\log x## we get ##G(\log ;x_0)= \frac{1}{x_0} = \frac{1}{\exp f(x_0)}##. So a logarithmic growth is reciprocal to the value of the "instant" point.

However, "growth rate" should actually be the second derivative (which of course doesn't affect the result in case of the exponential function, only the proportion factor).

scottdave
Homework Helper
In my experience, Logarithms are used as a way of expressing some huge (or tiny) values as something easy to manage. For example, decibels, Richter scale, pH. These are all units which the raw value (or ratio) would be very large magnitudes of change, can be expressed in 1 or 2 digit values.

As a growth function, I do not know of something that occurs which grows logarithmicly, but they are correct about it being slow. For example, the derivative (slope) of the LN(x) function is 1/x (where LN is the natural log). So when x is 10, the rate of change is 0.1, but when x is 1000, the rate has slowed to 0.001 and it gets slower as it increases (but yet it still is increasing). As it approaches infinity, the rate of change will approach zero (but not quite).

symbolipoint
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