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Exponents and Logarithms

In general, an exponential function is of the

form f(x) ax where a gt 0 is a constant

A logarithmic function is of the form f(x)

logax where a gt 0 is a constant.

Exponential Functions

- The graph of an exponential function f(x) ax

where a gt 1 looks like

e.g. This is y 2x

All exponential functions pass through the point

(0,1) and are asymptotic to the x-axis.

Graphs of Exponential Functions

The yex Function

Exponential functions where 0 lt a lt 1

- Recall from earlier discussions how f(-x)

differs from f(x). - So, if f(x) 2x, then f(-x) 2-x

(2-1)x or ( ½ )x whose graph looks like

Inverse Functions

- 1-1 Functions f(x) f(y) --gt x y
- horizontal line test

f(x) x3 2 Show this function to be 1-1 and

find its inverse.

The graph shows that the function is 1-1 since it

passes the horizontal line test.

Finding the Inverse Function

- 1. Write the function solved for y as a

function of x. - 2. Interchange the variables x and y.
- 3. Solve the new equation for y.
- 1. f(x) x3 2 --gt y x3 2
- 2. x y3 2
- 3. x - 2 y3 or y (x - 2)(1/3)

y x3 2 blue y (x - 2)(1/3) red

y x

Find the inverse for f(x)

y

x

x2 -1 - y y -x2 - 1 x ? 0

Find and graph the inverse function using

parametric equations.

Original function x(t) t

Inverse function y(t) t

The Graph

Original Function

Inverse Function

Logarithmic Functions

- Logarithms are inverse functions for exponential

functions - f(x) 2x f-1(x) log2x

Definition of a Logarithm

- loga x y iff ay x

Common Logs -- Base 10 Log 7 means log10

7 Natural Logs -- Base e ln 14.3 means

loge 14.3

Laws of Logarithms

- loga(x y) logax logay xgt0, ygt0
- log2(4 x 8) log24 log28 2 3 5
- loga(x/y) logax - logay
- log5(25/125) log525 - log5125 2 - 3 -1
- loga(xr) r logax where r is any Real no.
- log1021024 1024 log102 1024 x .3010
- 308.224

Examples

- log3 81

4

log 1000

3

6

log7 493

log2 ½

-1

ln e5

5

log5

1/3

log9 27

2/3

-3

log1/2 8

Example Given log 2 .3010, log 3

.4771, Find

- log 4
- log 8
- log 6
- log 9
- log 5
- log 12
- log 30
- log 3,000,000

- 0.6020
- 0.9030

Change of Base Formula

For example Find log795

Sketch the graph of y ln(x - 2) - 1

y ln(x - 2)

y ln(x - 2) - 1