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The natural logarithm, denoted as ln x, is the inverse of
the exponential function.
It actually comes up on its own in many contexts, also. It
has two important properties which can be deduced from the
two fundamental properties of the exponent
The definition of (ln x) can be phrased this way: it is the
power that you must raise e to in order to get x: e^(ln x)
= x.
We often raise other numbers, in particular 2 and 10 to powers
and can ask in general: what power must you raise z to in
order to get x? The answer is called the logarithm of x to
base z and is written as logzx. It is the inverse
function to z^x.
The two important properties alluded to above can be written
as
ln a * b = ln a + ln b.
and
logab *
logbc = logac for any a, b and c.
Exercises:
2.13 Deduce them (I must confess that I always get confused
in doing so by poor notation, but I am sure you, being younger
and smarter, can do it.) Solution
2.14 Deduce from these two equations that for log to any
base we have log a * b = log a + log b. Solution
The first property or rather the result in exercise 2.13 implies
that we can perform the multiplication a * b by taking the
logarithms of a and of b, adding them and then retrieving
a * b from its logarithm. Thus multiplication could be reduced
to addition and taking logarithms and 'antilogarithms'. Before
the advent of calculators, with which addition and multiplication
are equally difficult, this was an important use for logarithms,
and I recall as a high school student being forced to drill
doing this with log tables and antilog tables and of course
having to interpolate between values in those tables. It is
hard to imagine anything mathematical more tedious than this,
you lucky dogs!
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