Question

Difference between logarithmic growth and exponential growth

Answer 1

Exponential growth refers to a pattern of growth that the rate of increase is proportional to its current value. By definition, exponential growth is modeled by the equation:

xt=x0rtxt=x0rt

where x0x0 is the initial value at t=0,rt=0,ris the growth rate, tt is the time elapsed and xtxtis the value of xx at time tt. Notice that r<1r<1denotes a decay rate while r>1r>1denotes a growth rate.

A characteristic of an exponential function is that its rate of change is directly proportional to its rate of rate of change. Hence the rate of change will be ever-increasing with it increase in time. This can be see by this.

f(t)=x0rtf(t)=x0rt

f′(t)=x0lnr∗rtf′(t)=x0lnr∗rt

f′′(t)=x0(lnr)2∗rtf″(t)=x0(lnr)2∗rt

Logarithmic growth, however, is opposite in nature. By definition, it occurs when the rate of increase is inversely proportional to its currently value. It can be summarized by this equation:

xt=x0lntxt=x0lnt

The rate of change of a logarithmic growth is ever slowing down. This can be seen in the following:

f(t)=x0lntf(t)=x0lnt

f′(t)=x0tf′(t)=x0t

f′′(t)=−x02t2f″(t)=−x02t2

The first derivative of f(x)f(x)denotes the rate of change of x and it is a fraction, meaning that its rate of increase would become smaller and smaller until reaching 0 at infinity. Its second derivative is the rate of rate of change and its negative, showing that the rate of change decreases at a rate government by it.

In short, both growth would attend infinity when t attends infinity. However, exponential growth would grow first slowly and than rapidly increase in its speed. Logarithmic growth, in contrast, would initially grow quickly, then gradually slow down its growth rate.

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Answer 2

The exponential growth model shows a characteristic curve which is J-shaped while the logistic grown model shows a characteristic curve which is S-shaped.

The exponential growth model is applicable to any population which doesn’t have a limit for growth. The logistic growth model is applicable to any population which comes to a carrying capacity.

The exponential growth model typically results in an explosion of the population. The logistic growth model results in a relatively constant rate of population growth. This happens when the growth rate of the population arrives at its carrying capacity.

Exponential growth is ideal for populations that have unlimited resources and space – such as bacterial cultures. Logistic growth is more realistic and can be applied to different populations which exist in the planet.

The exponential growth model doesn’t have any upper limit. The logistic growth model has and upper limit, which is the carrying capacity.

Exponential growth happens when the rate of growth is in proportion to the existing amounts. This is also true for logistic growth but the difference is, it also includes competition and resources which are limited.