Logarithms are the "opposite"
of exponentials,
just as subtraction is the opposite of addition and division is the opposite
of multiplication. Logs "undo" exponentials. Technically speaking,
logs are the inverses
of exponentials.

In practical terms, I have
found it useful to think of logs in terms of The Relationship:

—The
Relationship—

y = b^{x}

..............is equivalent to...............
(means the exact same thing as)

log_{b}(y)
= x

On the left-hand side above
is the exponential statement "y
= b^{x}".
On the right-hand side above, "log_{b}(y)
= x"
is the equivalent logarithmic statement, which is pronounced "log-base-b
of y
equals x";
The value of the subscripted "b"
is "the base of the logarithm", just as b
is the base in the exponential expression "b^{x}".
And, just as the base b
in an exponential is always positive and not equal to 1,
so also the base b
for a logarithm is always positive and not equal to 1.
Whatever is inside the logarithm is called the "argument" of
the log. Note that the base in both the exponential equation and the log
equation (above) is "b",
but that the x
and y
switch sides when you switch between the two equations.

—The
Relationship Animated—

If you can remember this
relationship (that whatever had been the argument of the log becomes
the "equals" and whatever had been the "equals"
becomes the exponent in the exponential, and vice versa), then you shouldn't
have too much trouble with logarithms.

(I coined the term "The
Relationship" myself. You will not find it in your text, and
your teachers and tutors will have no idea what you're talking about if
you mention it to them. "The Relationship" is entirely non-standard
terminology. Why do I use it anyway? Because it works.)

By the way: If you noticed
that I switched the variables between the two boxes displaying "The
Relationship", you've got a sharp eye. I did that on purpose, to
stress that the point is not the variables themselves, but how they move.

Convert "6^{3}
= 216" to the equivalent logarithmic
expression.

To convert, the base
(that is, the 6)remains
the same, but the
3 and the 216
switch sides. This gives me: