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The Natural Base e

The Natural Base e 1.17 - Solution

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Recall that the general form of an exponential function is y=abx,y=ab^x, where a0a \neq 0 is a factor that can shrink, stretch, or reflect the function. The base bb must be positive. If a>0a>0 and b>1,b>1, then we have an exponential growth function and bb is called the growth factor.

On the other hand, if the base is a>0a>0 but 0<b<1,0<b<1, we have an exponential decay function and bb is known as the decay factor.

We can compare the given function y=13exy=\frac{1}{3}e^x to the general form of an exponential function to identify its parameters and determine if it represents exponential growth or exponential decay. y=abxy=13exa=13b=e\begin{gathered} y={\color{#0000FF}{a}}{\color{#009600}{b}}^x \qquad y={\color{#0000FF}{\frac{1}{3}}}{\color{#009600}{e}}^x\\[0.5em] \Downarrow\\[0.5em] {\color{#0000FF}{a}}={\color{#0000FF}{\frac{1}{3}}} \qquad {\color{#009600}{b}}={\color{#009600}{e}} \end{gathered} Notice that a=13a = \frac{1}{3} is positive. Furthermore, the base is the natural base ee, which has an approximate value e2.718281e \approx 2.718281\ldots and is greater than 1.1. Hence, the function y=13exy=\frac{1}{3}e^x represents exponential growth.