Econ 407:

Log and Exponential Functions.

Examples of Exponential Functions and their graphs

> y:=a^x;

The simplest exponential function--where a is a constant and x is a variable.

> y:=2^x;

> plot(y,x=0..2);

> Wpop:=6^x;

> plot(Wpop,x=0..2.3);

> plot({0.3^x,0.4^x,0.5^x},x=-2.0..3);

> plot({3^(-x),4^(-x),5^(-x)}, x=-3..3);

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> y:=exp(0.5*x);

> eval(y,x=2);

> y:=exp(x);

> dy:=diff(y,x);

> plot(y,x=-2..2);

> plot({y,dy},x=-2..2);

An important exponential function is y=e^x, where e is the base of the natural log. One of the interesting properties of this function is that its first, second, third derivatives are identical to the original function.

> y:=exp(x);

> y:=exp(2*x);

> dy:=diff(y,x);

> plot({y,dy},x=-2..2);

> y:=exp(5*x^2);

> dy:=diff(y,x);

> y:=exp(5*x^5);

> dy:=diff(y,x);

> y:=exp(5*x^(-2));

> dy:=diff(y,x);

Here is an example of a demand and supply function which depend on the natural log of price. Note that in such a function, the coefficient of price (i.e. the number multiplied by log(p) is a measure of elastcity of demand or supply.

> qd:=120-8*log(p);

> qs:=50*log(p)-60.0;

> eval(qd,p=19.0);

> eval(qs,p=19.0);

> solve(qd=qs,p);

> eval(qd,p=22.27462807);

> eval(qs,p=22.27462807);

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