Exponential function

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The exponential function is the function $f(x) = e^x$ , exponentiation by e. It is a very important function in analysis, both real and complex.

General Info and Definitions

Exponential functions are functions that grows or decays at a constant percent rate.

Exponential functions that result in an increase of y is called an exponential growth.

Exponential functions that result in an decrease of y is called an exponential decay.

An exponential growth graph looks like:

An exponential decay graph looks like:

Exponential functions are in one of three forms.

$f\left( x \right) = ab^x$ , where b is the % change written in decimals

$f\left( x \right) = ae^k$ , where e is the irrational constant 2.71828182846....

$f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}}$ or $f\left( x \right) = a\left( 2 \right)^{{x \over d}}$ , where h is the half-life (for decay), or d is the doubling time (for growth).

Whether an exponential function shows growth or decay depends upon the value of its b value.

If $b > 1$ , then the funciton will show growth.

If $0 < b < 1$ , then the function will show decay.

Solving Exponential Equations

There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using logarithms.

Example: Solve $56 = 12\left( {1.24976} \right)^x$

Graphically:

Graph both equations and find the intersection.

Algebraicly:

There, I will use natural logarithms. The same opperation can also be done with common logarithms.

$56 = 12\left( {1.24976} \right)^x$

${{56} \over {12}} = \left( {1.24976} \right)^x$

$\ln \left( {{{56} \over {12}}} \right) = x\ln \left( {1.24976} \right)$

$x = {{\ln \left( {{{56} \over {12}}} \right)} \over {\ln \left( {1.24976} \right)}}$

$x \approx 6.9093$

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Exponential function

From AoPSWiki

General Info and Definitions

Solving Exponential Equations