5.1
Exponential functions
An exponential (or power) function is of the form
a
is known as the base
x
is known as the exponent, power or index.
Remember the following rules for indices:
1.
2.
3.
4.
5.
6.
a
0
1
q
2
a
p
a
p
q
1
a
p
a
p
1
a
p
2
q
a
pq
a
p
a
q
a
p
q
a
p
a
q
a
p
q
1
a
1
2
.
y
a
x
.
5
Exponential and Logarithmic Functions
109
108
5
Exponential and Logarithmic
Functions
1
In this chapter we will meet logarithms,
which have many important applications,
particularly in the field of natural science.
Logarithms were invented by John Napier
as an aid to computation in the 16th
century.
John Napier was born in Edinburgh,
Scotland, in 1550. Few records exist about
John Napier’s early life, but it is known that
he was educated at St Andrews University,
beginning in 1563 at the age of 13.
However, it appears that he did not graduate
from the university as his name does not
appear on any subsequent pass lists.The
assumption is that Napier left to study in
Europe.There is no record of where he
went, but the University of Paris is likely,
and there are also indications that he spent
time in Italy and the Netherlands.
While at St.Andrews University, Napier became very interested in theology and he
took part in the religious controversies of the time. He was a devout Protestant, and
his most important work, the
Plaine Discovery of the Whole Revelation of St.John
was published
in 1593.
It is not clear where Napier learned mathematics, but it remained a hobby of his, with
him saying that he often found it hard to find the time to work on it alongside his
work on theology. He is best remembered for his invention of logarithms, which
were used by Kepler, whose work was the basis for Newton’s theory of gravitation.
However his mathematics went beyond this and he also worked on exponential
expressions for trigonometric functions, the decimal notation for fractions, a
mnemonic for formulae used in solving spherical triangles, and “Napier’s analogies”,
two formulae used in solving spherical triangles. He was also the inventor of
“Napier’s bones”, used for mechanically multiplying, dividing and taking square and
cube roots. Napier also found exponential epressions for trignometric functions, and
introduced the decimal notation for fractions.
We can still sympathize with his sentiments today, when in the preface to the
Mirifici
logarithmorum canonis descriptio
, Napier says he hopes that his “logarithms will save
calculators much time and free them from the
slippery errors
of calculations”.
John Napier
Example
Simplify
x
1
5
x
2
5
5
2
x
3
x
3
5
x
3
5
x
0
1
x
1
5
x
2
5
5
2
x
3
.
Example
Evaluate
without a calculator.
8
2
3
1
3
2
8
2
1
2
2
1
4
8
2
3
Graphing exponential functions
Consider the function
y
2
x
.
x
0
1
2
3
4
5
y
1
2
4
8
16
32
1
2
1
4
1
2
The
y
-values double for
every integral increase of
x
.

Now
All exponential graphs of the form
can
be expressed in this way, and from our knowledge of transformations of functions this is
actually a reflection in the
y
-axis.