EXPONENTIAL FUNCTIONS
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Exponential Functions

First review: Exponents

The function

y = e^x is the exponential function.

e is the constant 2.71 and is called the "natural number e."

Its graph looks like:

Notes:

The base e is commonly used in advanced Mathematics and Physics. Another common base is 10. PH values in Chemistry are based on the exponents of base 10 numbers. Here's a graph of 10^x and how it compares to the e^x.

Notes:

1) since 10 is greater than 2.71 the base 10 exponential will increase more rapidly as x increases above 0 and the base 10 exponential will decrease more rapidly as x decrease below 0. Both intersect at (0, 1). Why? True for all exponential functions? No, only those with no leading coefficients, e.g., 5e^x crosses the Y axis at y = 5. (5e⁰ = 5 • 1 = 5)

The graphs of e^-x and 10^-x:

Notes:

1) The previous functions increase with no limit, these decrease toward zero as a limit, as x increases.

2) The x axis (x=0) is the asymptote of exponential functions.

The plot of a negative exponential function is the plot of its positive function translated across the x axis.

For example: –e^x

Manipulating Exponentials

First:

The rules for exponents apply to exponential functions.

For example:

a bit more complicated example:

e^x/3 • e^-2x = e ^{(1 – 2•3)/3} x = e ^{– 5/3 x} = (e ^-5x)^1/3 = (1/e^5x)^1/3

so, we have the cube root of the reciprocal of e^5x

       Now, solving exponential equations hinges on the fact that, if two exponentials are equal, then the exponents must be equal.

For example:

e ^2x = e ^{3x - 1}

so, 2x = 3x -1 solving for x we get x = 1

and:

e^{x - 1} = e ⁴⁵

here x - 1 = 45, so x = 44

Without using logarithms, these are the only types of exponential equations we can solve.

The next example is more complex, it requires the quadratic equation to solve.

An exponential function can have any base, even negative. However we usually limit the bases to be greater than zero.

The standard forms of an exponential equation are:

y : dependent variable

A : constant

b : base ( > 0 )

k : constant

x : independent variable (Real)

C : log_b A

Examples: Write the following equations in standard form: