- logaa1, because a^1 a. For example, in the equation 3^1 3, the base is 3
and the exponent is 1; the result is 3, with the corresponding logarithmic equa-
tion being log 33 1. - logaaxx, because axax. For example, 3^4 34 , with the base as 3. The loga-
rithmic equation becomes log 334 4.
What is an exponential function?
Along with exponents come exponential functions, or the relationship between values
of a variable and the numbers formed by raising some positive number to the power of
those values. In functional notation, an exponential function is written f(x) ax, in
which ais a positive number; for example, the function f(x) 2 xis an exponential
function.
In logarithmic terms, an exponential function is most commonly written as exp
(x) or ex, in which eis called the base of the natural logarithm. These types of func-
tions are usually shown on a graph. (For more information about graphs, see “Geome-
try and Trigonometry.”) As a function of the real variable x,the resulting graph of exis
always positive, or above the xaxis and increasing from left to right. Although the line
of such a function never touches the xaxis, it gets very close to it.
What does erepresentin terms of logarithms?
No, eis not the code name in a James Bond movie. When talking about logarithms
(or logs), in the majority of mathematical circles, it means the base of the natural
logarithm. It is yet another irrational, transcendental number (such as pi, or π) that
has a plethora of names: It has been called everything from the logarithmic constant
and Napier’s number to Euler’s constant and the natural logarithmic base. One of the
best ways to define eis to use the expression (1 x)(1/x); eis the number that this
expression approaches as xgets smaller and smaller. Substituting in values for x
gives one a better idea: if x1, the result is 2; if x0.5, the result is 2.25; when x
0.25, the result is 2.4414 ...; if x0.125, the result is 2.56578 ...; if x0.0625, the
result is 2.63792 ...; and so on. This is why approximations are often used in solving
equations using e. 147
ALGEBRA
What are some examples of logarithm use?
L
ogarithms are used in many areas of science and engineering, especially in
those areas in which quantities vary over a large range. For example, the
decibel scale for the loudness of sound and the astronomical scale of stellar
brightness are both logarithmic scales.