Lovelace, Augusta Ada Byron 321

)

1

–––

xlnb

dy

––

dx

1

–

x

dy

––

dx

1

–

x

1

–

t

1

–

x

d

––

dx

p

–

q

p

–

q

p

–

q

4. If f(x) is a logarithmic function with f(b) = 1, then

f(x) = logbx.

(For any positive integer n, we have f(xn) = f(x) + f(x)

+…+ f(x) = nf(x).Also, .

For a fraction , we have qf(x) = f

(

(x)q

)

= f(xp)

= pf (x), which shows f(xr) = rf(x) for any rational

number r. Since every irrational number is a

LIMIT

of

rationals, this property also holds for irrational num-

bers. In particular, for any value x, f(bx) = xf(b) = x.

That is, fis indeed the function that extracts the expo-

nent of a given power of b.)

If a logarithmic function satisfies f(e) = 1, then it is

called the logarithmic function. It is the function given

by the natural logarithm: f(x) = ln x. Its derivative is

given by (ln x) = , and consequently ln x=

∫

xdt.

Thus the logarithmic function can alternatively be

defined as the area under the y= curve. (See

E

.)

The derivative of a logarithmic function to an arbi-

trary base bcan be computed via implicit differentiation

after first unraveling and then reapplying a logarithm.

Specifically, if y= logbx, then by= x. Applying the natu-

ral logarithm yields ylnb= lnx. Differentiating, we see

that ln b= , and so, = . Thus we have:

See also

HOMOMORPHISM

;

LOGARITHMIC SCALE

;

SLIDE RULE

.

logarithmic scale A line marked with distances whose

LOGARITHM

s are the actual distances along the line is

said to be in logarithmic scale. For example, in a base-10

logarithmic scale, if 10 is the label 1 in. along the line,

each successive inch will be marked 100, 1,000, etc.

(The position labeled 1,000, for example, is physically

log10 1,000 = 3 in. along the line.) If a set of coordinate

axes are both in logarithmic scale, then the plot of a

curve y= xn, for example, will appear as the plot of

log y= log(xn) = n log x, that is, as though the variables

in question are logyand logx. In this setting, the graph

appears as a straight line of

SLOPE

n. Scientists often plot

graphs on log-log graph paper and measure slope to find

the value of nthat best fits the data.

The term logarithmic scale is also used to describe

any quantity that is typically measured in terms of loga-

rithms. In 1935 American seismologist Charles Richter

set up the Richter scale to describe the intensity of

earthquakes. The scale he devised is logarithmic, base

10, meaning that an earthquake measuring 6 on his

scale, for example, is 106, or 1 million, times stronger

than the small quakes that he set at value zero. The pH

measurement of acidity or alkalinity used in chemistry is

also logarithmic, as is the unit

DECIBEL

used to measure

sound intensity.

See also

SLIDE RULE

.

logistic growth See

POPULATION MODELS

.

long division See

BASE OF A NUMBER SYSTEM

.

long radius The distance from the center of a regular

POLYGON

to one of its vertices is called the long radius

of the polygon. This quantity is also the radius of

CIR

-

CUMCIRCLE

of the polygon. If the polygon has nsides,

each 1 unit in length, then an exercise in

TRIGONOME

-

TRY

shows that the long radius has value

An analog of

PI

for a regular polygon is the ratio of

its

PERIMETER

to twice the length of it long radius

(which equals the diameter of the polygon if nis even).

This quantity approaches the value as πas nbecomes

large. This follows from the fact that, according to the

SQUEEZE RULE

, approaches the value 1 as x,

measured in radians, becomes small.

See also

APOTHEM

;

RADIAN MEASURE

.

Lovelace, Augusta Ada Byron (1815–1852) British

Computation Born on December 10, 1815, in Pic-

cadilly, England, Augusta Ada Byron Lovelace is remem-

bered as one of the first people to write a set of

instructions for a computing machine. As an assistant to

sin(x)

––

x