Table of Contents

**What is an Exponential Function?**

Exponential function is one of type of function. It is non algebraic (transcendental) function which cannot operate with addition, subtraction, multiplication, division of difference variables raised to some non-negative integer power.

Exponential function is not only in mathematics, but also in physics as the application.

**Exponential Function Equation**

The general form of exponential function equation is

*f(x) = a*^{x}

note:

*a*is base. It is any real numbers that greater than 0.- x is variable of the power (exponent). It is any real numbers.
- exponential function has real numbers as the domain and positive real numbers as the range.

**Exponential Function Graph**

Graphing exponential function is like graphing other function. The steps are:

- Make a list and determine the value of variable start from simple number (0,1,2,…).
- Calculate the value of the exponential function by substituting the value into the function.
- Draw the graph by using the point (coordinate)
- Linking the points.

**Properties of exponential function**

There are properties of exponential function f(x) = *a*^{x}:

- Graph of the function is always contain point (0,1). It is mean that if x = 0 then f(x) =
*a*^{0 }= 1. - For every number of
*a*, it is always*a*^{x}≥ 0. - The graph of
*a*^{x}will decrease if 0 <*a*< 1. - The graph will increase if
*a*^{x}> 1 - If
*a*^{x}=*a*^{y}then x = y.

**Inverse**

Exponential function has inverse. The inverse is logarithmic function.

**f(x) = a**** ^{x}** then the inverse is

**log**

_{a}**(x)**

Besides that, sometimes it is also use natural logarithmic function (ln)

**Natural Exponential Function**

Exponential function is not only having general form equation, it is also specific form that is called natural exponential function.

f(x) = e^{x}

note:

e = 2.71828…

**Exponential Function Examples**

- Sketch the graph of f(x) = 2
^{x}.

Making list to determine the point (coordinate) of the graph.

X | f(x) = 2^{x} | Coordinate |

0 | 2^{0} = 1 | (0,1) |

1 | 2^{1} = 2 | (1,2) |

2 | 2^{2} = 4 | (2,4) |

3 | 2^{3} = 6 | (3,6) |

4 | 2^{4} = 8 | (4,8) |

Then draw the graph appropriate with the point.

2. Solving each problems of exponential function

a. 3^{2x} = 3^{1-3x}

b. 7^{1-x} = 4^{3x+1}

c. e^{4-7x} + 11 = 20

a. 3^{2x} = 3^{1-3x}

Then

2x = 1 – 3x

2x + 3x = 1

5x = 1

x = 1/5

b. 7^{1-x} = 4^{3x+1}

Because of the base are not same, it uses (natural) logarithm concept.

ln7^{1-x} = ln4^{3x+1}^{}

then using logarithm properties that log_{a}x^{r} = r.log_{a}x, it will get

(1-x) ln7 = (3x+1) ln4

ln7 – x.ln7 = 3x.ln4 + ln4

ln7 – ln4 = 3x.ln4 + x.ln7

ln7 – ln4 = (3.ln4 + ln7)x

x = (ln7 – ln4)/(3.ln4 + ln7)

= 0,09166..

c. e^{4-7x} + 11 = 20

e^{4-7x} = 20 – 11

e^{4-7x} = 9

then using (natural) logarithm, we get:

lne^{4-7x} = ln9

use logarithm properties

lne^{f(x)} = f(x)

then

lne^{4-7x} = ln9

4 – 7x = ln9

7x = 4 – ln9

x = (4 – ln9)/7

= 0.2575…