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### Definition of Logarithm

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Date: 04/16/2001 at 08:20:23
From: Hongki, Woo
Subject: The definition of logarithm, about base

In the definition of logarithm, there are some conditions for the
base. a^x = b iff x = log_a(b) where a > 0, a <> 1 (a is not equal to
1), and b > 0. Why stipulate these conditions? If they are not
assumed, are there any problems or contradictions that arise? I want
to know why mathematicians defined logarithm this way.
```

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Date: 04/16/2001 at 12:50:57
From: Doctor Peterson
Subject: Re: The definition of logarithm, about base

Thanks for writing to Dr. Math.

Let's just think about what happens if we violate any of these
conditions.

First, if b <= 0, then we are trying to solve:

a^x <= 0

You will find that there is no solution; x must be "negatively
infinite" even to produce 0, and there is no real number x for which
a^x < 0 if a > 0.

Next, if a = 1, we are trying to solve:

1^x = b

Since 1^x = 1 for all x, this has no solution unless b = 1. So it
makes no sense to talk about a logarithm with base 1.

Now, if a = 0, we have:

0^x = b

and again, there is no solution unless b = 0. This is likewise
useless.

Finally, if a < 0, things get tricky. We can define integral powers of
a (which will be positive for even x and negative for odd x); but what
about fractional powers? a^(1/2) is undefined (or rather, imaginary)
when a is negative. If the powers are not defined for a negative base,
then logarithms are not defined either!

As you can see, the restrictions you asked about merely keep us from
talking about logarithms when they don't make any sense.

BUT ...

In the first and last cases, you'll notice I mentioned real and
imaginary numbers. The fact is, your definition is valid only when we
are considering only real numbers. We can extend the logarithm to
apply to complex numbers, and then some of these restrictions can be
relaxed. You can read about complex logs and some of the complications
they introduce in the Dr. Math archives:

The Log of a Negative Number
http://mathforum.org/dr.math/problems/witty3.27.98.html

Log of Complex Number
http://mathforum.org/dr.math/problems/langlands.9.15.96.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
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