For question involving exponential functions and questions on exponential growth or decay.

The exponential function is one of the most important functions in mathematics (though it would have to admit that the linear function ranks even higher in importance).

Definition:If $~b~$ be any number such that $~b\gt 0~$ and $~b\neq 1~$ then an exponential function is a function in the form,$$f(x)=a~b^x$$ where $~b~$ is called thebase, the exponent,$~x~$ can be any real number and $~a\neq0~$.

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**Properties:**

- The graph of $~f(x)~$ will always contain the point $~(0,1)~$. Or put another way, $~f(0)=1~$ regardless of the value of $~b~$.
- For every possible $~b~$we have $~b^x\gt 0~$. Note that this implies that $~b^x\neq 0~$.
- If $~0\lt b\lt 1~$then the graph of $~b^x~$ will decrease as we move from left to right. Check out the graph of $~\left(\frac{1}{2}\right)^x~$ above for verification of this property.
- If $~b\gt 1~$ then the graph of $~b^x~$ will increase as we move from left to right. Check out the graph of $~2^x~$ above for verification of this property.
- If $~b^x=b^y~$, then $~x=y~$.

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**The Natural Exponential Function:**
In mathematics, the **natural** exponential function is $$f(x)=e^x~,$$ where $e$ is Euler's number.

**Note:** $f(x)=e^x~$ is a special exponential function. In fact this is so special that for many people this is *THE exponential function*.

**Applications:**

Exponential functions are solutions to the simplest types of dynamical systems. It is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. Exponential decay occurs in the same way when the growth rate is negative.

**References:**