Exponential Functions · Exercise 68

In an exponential decay function, the base of the exponent is a value between 0 and 1. Thus, for some number $b>1,$ the exponential decay function can be written as $f\left(x\right)=a\cdot {\left(\frac{1}{b}\right)}^x.$ Use this formula, along with the fact that $b=e^n,$ to show that an exponential decay function takes the form $f\left(x\right)=a{\left(e\right)}^{-nx}$ for some positive number $n$.

OpenStax, Algebra and Trigonometry 2e · §6.1 Exponential Functions · Exercise 68