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Exponential function

An exponential function is a function of the form $$$ y = \Tblue{k} \Torange{a}^{x}.$$$ The number $$\Torange{a}$$ is positive.

The computation without a calculator is difficult for a general $$x$$. However, it can be done when $$x$$ is rational.

  • When $$x$$ is an integer: $$y$$ is the usual $$\Torange{a}$$ to the power of $$x$$. $$$ 2^{\Tred{1}} = 2,\; 2^{\Tred{2}} = 2\times2 = 4,\; 2^{\Tred{3}} = 2^2\times 2 = 8,\; 2^{\Tred{4}} = 16.$$$
  • When $$x$$ is the reciprocal of an integer: $$y$$ is the $$x$$-th root of $$\Torange{a}$$, i.e. the number that is $$\Torange{a}$$ when raised to the power $$x$$ $$$ 2^{1/\Tred{2}} = \sqrt{2} \simeq 1.41,\; 2^{1/\Tred{3}} = \sqrt[\Tred{3}]{2}\simeq 1.26,\; 2^{1/\Tred{4}} = \sqrt[\Tred{4}]{2}\simeq 1.19.$$$
  • When $$x$$ is rational ($$p/q$$): $$y$$ is the $$q$$-th root of $$\Torange{a}$$ raised to the power $$p$$ $$$ 2^{\Tblue{3}/\Tred{2}} = (\sqrt{2})^\Tblue{3} \simeq (1.41)^\Tblue{3}\simeq 1.58,\; 2^{\Tblue{2}/\Tred{3}} = (\sqrt[\Tred{3}]{2})^\Tblue{2}\simeq 2.83$$$
Graph of the exponential function $$y = 2^x$$
Graph of the exponential function $$y = 2^x$$