Carisa
Lindsay

EMAT
6690

Logarithmic
Functions

Logarithmic
functions are essentially just inverses of exponential functions. This is how we are often taught in school,
but there is seldom any further investigation as to why this is true. Students are simply told that this is how it
is.

As
you can see in the above graphic, logarithms are truly inverses of exponential
functions since it is a reflection over the line y=x.

Logarithmic
functions behave in similar ways as our more familiar functions do- we can
stretch, compress, reflect, and translate them.

For
instance,

y=2*log
x

will
yield a result of the parent function vertically stretched by a factor of
2.

We
will see a similar result for reflections and translations:

y=-log
x

y=log(x+3)

__The domain of Logarithmic Functions__

As
you can see on the following graphs, which has not been horizontally
translated, the lowest defined value of the logarithmic function is 0.

The
domain will continue onto infinity (we can verify this through testing various
points and make assumptions about the end behavior).

So
far, we have focused primarily on logarithms with a base of 10. What happens if we change the base? How are the graphs similar? Different?

If
we consider simple logarithms free of transformations, we can clearly see the
effect the base will have on the graph.
Essentially, as we increase the base number, the less steep it becomes. It appears that the growth changes less
steadily as the base increases. We also
notice that all of the logarithms have a common point of intersection, (1,0). This is not
surprising since the log of 1 is always zero.

Click here to see an animation.

You
may wonder, what is happening for values less than one? It seems to be inverting itself as the base
becomes less than one. Let’s take a
closer look.

WE
need the definition of a logarithm to help us explain why this occurs. Remember, logarithms are simply inverses of
exponential functions. The logarithms we
have looked at so far have looked like exponential growth. With rational base values, we seem to
experience something like exponential decay.
Let’s convert our logarithms to exponential form to see if they truly
are examples of exponential decay.

*y* =log_{0.75 }(*x*) à
(0.75)* ^{y}*
=

Since
our base value is less than one, we know the graph exhibits exponential
decay. The base of the logarithm is
essentially the base of the exponent, so all of our rational bases will be
examples of exponential decay. For the same reason, our bases greater than one will have an
exponential growth pattern.

__What about values less than one? __

For
base values less than one, we have undefined values.

__What are logarithms useful for?__

Logarithmic
functions are especially useful to us to approximate and apply to fields other
than mathematics.

Logarithms
are mostly taught through solving interest problems as a means of solving an
exponential equation. They can also be
used for solving various mortgage problems (since they are similar to interest
problems).

We
can also use logarithms to solve population problems and predict population
growth as well as radioactive decay.

Logarithms
are also used with earthquake magnitude.
The Richter scale is based on base 10, so we can use logarithms to solve
for unknown quantities.

Chemists
also use logarithms when discussing the pH of various substances when
determining the acidity or how basic the substance is.

Logarithms
can also be used for determining the intensity of sound (volume). The decibel scale is essentially exponential
and therefore logarithmic also.

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