Exponential Function

A function of the form f(x)=ax{f{{\left({x}\right)}}}={{a}}^{{x}}, where a>0{a}>{0}, is called an exponential function.

Do not confuse it with the power function f(x)=xa{f{{\left({x}\right)}}}={{x}}^{{a}} in which the variable is the base.

The domain of the exponential function is (,){\left(-\infty,\infty\right)}, and the range is (0,){\left({0},\infty\right)}, provided a1{a}\ne{1}.

If a=1{a}={1}, we have that f(x)=1x=1{f{{\left({x}\right)}}}={{1}}^{{x}}={1}.

Exponential functions are useful for modeling many natural phenomena such as population growth (if a>1{a}>{1}) and radioactive decay (if 0<a<1{0}<{a}<{1}).

Let's see what is meant under f(x)=ax{f{{\left({x}\right)}}}={{a}}^{{x}}.

If x=n{x}={n}, where n{n} is a postive integer, we have that an=(aaa)n{{a}}^{{n}}=\underbrace{\left({a}\cdot{a}\cdot\ldots\cdot{a}\right)}_{{{n}}}.

If x=0{x}={0}, it can be stated that a0=1{{a}}^{{0}}={1}, and if x=n{x}=-{n}, where n{n} is a positive integer, we have that an=1an{{a}}^{{-{n}}}=\frac{{1}}{{{{a}}^{{n}}}}.

If x{x} is a rational number, x=pq{x}=\frac{{p}}{{q}}, where p{p} and q{q} are integers, it can be stated that ax=apq=apq=(aq)p{{a}}^{{x}}={{a}}^{{\frac{{p}}{{q}}}}={\sqrt[{{q}}]{{{{a}}^{{p}}}}}={{\left({\sqrt[{{q}}]{{a}}}\right)}}^{{p}}.

There are three kinds of exponential functions:

  1. If 0<a<1{0}<{a}<{1}, the exponential function decreases.
  2. If a=1{a}={1}, the exponential function is constant.
  3. If a>1{a}>{1}, it increases (the bigger a{a}, the more rapidly it increases).

exponential function

Properties of exponents.

If a{a} and b{b} are positive numbers, and x{x} and y{y} are any real numbers, it can be stated that:

  1. ax+y=axay{{a}}^{{{x}+{y}}}={{a}}^{{x}}{{a}}^{{y}}
  2. axy=axay{{a}}^{{{x}-{y}}}=\frac{{{{a}}^{{x}}}}{{{{a}}^{{y}}}}
  3. (ax)y=axy{{\left({{a}}^{{x}}\right)}}^{{{y}}}={{a}}^{{{x}{y}}}
  4. (ab)x=axbx{{\left({a}{b}\right)}}^{{x}}={{a}}^{{x}}{{b}}^{{x}}

Applications of exponential functions.

As stated above, exponential functions are widely used in growth/decay problems.

Example 1. Suppose that the initial population of some bacteria is 100. It is known that every 1 hour the population triples. Find the population after 10 hours.

If the number of bacteria at a time t{t} is p(t){p}{\left({t}\right)}, where t{t} is measured in hours, and p(0)=100{p}{\left({0}\right)}={100} is the initial population, we have that

p(1)=3p(0)=300{p}{\left({1}\right)}={3}{p}{\left({0}\right)}={300}

p(2)=3p(1)=33p(0)=32p(0)=900{p}{\left({2}\right)}={3}{p}{\left({1}\right)}={3}\cdot{3}\cdot{p}{\left({0}\right)}={{3}}^{{2}}{p}{\left({0}\right)}={900}

p(3)=3p(2)=333p(0)=33p(0)=2700{p}{\left({3}\right)}={3}{p}{\left({2}\right)}={3}\cdot{3}\cdot{3}\cdot{p}{\left({0}\right)}={{3}}^{{3}}{p}{\left({0}\right)}={2700}

p(4)=3p(3)=3333p(0)=34p(0)=8100{p}{\left({4}\right)}={3}{p}{\left({3}\right)}={3}\cdot{3}\cdot{3}\cdot{3}\cdot{p}{\left({0}\right)}={{3}}^{{4}}{p}{\left({0}\right)}={8100}.

Do you see the pattern? It seems that p(t)=1003n{p}{\left({t}\right)}={100}\cdot{{3}}^{{n}}.

So, p(10)=100310=5904900{p}{\left({10}\right)}={100}\cdot{{3}}^{{{10}}}={5904900}.

Now, let's proceed to another example.

Example 2. Suppose that the half-life of some radioactive element is 3 years (recall that the half-life is the amount of time needed to disintegrate a half of any quantity initially presented). Suppose that the initially presented mass is 1000 mg. Find the mass after 10 years.

If m(t){m}{\left({t}\right)} is the mass of the element at any time t{t} measured in years, we have that

m(0)=1000{m}{\left({0}\right)}={1000}

m(3)=12m(0)=500{m}{\left({3}\right)}=\frac{{1}}{{2}}{m}{\left({0}\right)}={500}

m(6)=12m(3)=122m(0)=250{m}{\left({6}\right)}=\frac{{1}}{{2}}{m}{\left({3}\right)}=\frac{{1}}{{{2}}^{{2}}}{m}{\left({0}\right)}={250}

From this pattern, we can conclude that

m(t)=12t31000=10002t3{m}{\left({t}\right)}=\frac{{1}}{{{{2}}^{{\frac{{t}}{{3}}}}}}\cdot{1000}={1000}\cdot{{2}}^{{-\frac{{t}}{{3}}}}.

So, in 10 years, there will be m(10)=1000210399.21{m}{\left({10}\right)}={1000}\cdot{{2}}^{{-\frac{{10}}{{3}}}}\approx{99.21} mg.