Function and Relation Library

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Function and Relation Library

LOGARITHMIC FUNCTIONS

y = log_b(x), where b is a positive number
opposite function: y = - log_b(x)
reciprocal function: y = 1/log_b(x)
inverse function: y = b^x
slope: y = (1/log_e(b))(1/x) or y = (log_b(e))(1/x)
ex. y = log₂(x)

Log, or logarithmic, functions undo or are the inverse function of power or exponential functions. The expression log_bx means the power to which b must be raised to in order to get x.

Visit the Interactive Sketch Pad Material
On Log Functions

logs -- Multi page: 2 logs, log w/exponential

See below how the x and y values for the log and exponential are interchanged.

a logrithm

x y=log₂(x)

1 0

2 1/2

2 1

4 2

8 3

an exponential

x y=2^x

0 1

1/2 2

1 2

2 4

3 8

Another way to look at the inverse relationship is to see how one function is the mirror image of the other -- x exchanges with y -- for the natural log and the exponential with base e. You may need to hit REFRESH or RELOAD on your browser in order for the animation to be visible.

The slope of log_bx is (log_be)(1/x) where x is greater than zero.

The inverse function is y = b^x .

Though any positive base may be used for a logarithm, the bases 10 and e are those most frequently used. Log₁₀ x is called the common log. Log_e x or ln x is called the natural log.

If a log with a base other than 10 or e is needed, it may be estimated or the following statement may be employed, often where c is either 10 or e.

log_ba =(log_ca)/(log_cb)

THE NATURAL LOGARITHM

y = ln(x)
opposite function: y = - ln(x)
reciprocal function: y = 1/ln(x)
inverse function: y = e^x
slope: y = 1/x

The natural log function is the inverse functions of the exponential function, e^x. The expression ln x means the power to which e must be raised to in order to get x. For example: Log_e x has values of 0, 1, 2, 3, and 1/2 for x values of 1, e, e², e³, and e^.5.

The slope of ln x is 1/x where x is greater than zero. The inverse function is y = e^x .

This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).

You are hereby granted permission to make ONE printed copy of this page and its picture(s) for your PERSONAL and not-for-profit use. YOU MAY NOT MAKE ANY ADDITIONAL COPIES OF THIS PAGE, ITS PICTURE(S), ITS SOUND CLIP(S), OR ITS ANIMATED GIFS WITHOUT PERMISSION.

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