Carbon-13 dating does indeed need corrections for the level of
solar activity, but that's a bit of an exception. The corrections
don't have *anything* to do with how fast the Carbon-13 decays,
though: they relate to how much C-13 is in the atmosphere. The
way it works is like this: Carbon-13 decays in about 5000 years.
Why do we still have some around then? That's because it's
constantly being made as cosmic rays hit the upper atmosphere of
the earth. Now, as a tree grows, it incorporates carbon from the
environment, including C-13. When the tree dies, the
incorporation pretty much stops. So, once the tree dies, no more
C-13 comes in, and whatever is there continues to decay. Thus,
some wood with a lot of C-13 is new because the C-13 hasn't had
time to decay. Wood with very little C-13 is old, because it has
been dead (i.e. disconnected from the environment) for long
enough for most of the C-13 to decay. So, how does solar activity
come into the picture? If the sun is inactive, more cosmic rays
come in from outer space and hit the Earth's atmosphere. Thus
more C-13 is made. Thus, a tree growing in a time with less solar
activity incorporates more C-13, and thus when you do C-13 dating
of it, it seems to be newer than it really is. The C-13 always
decays at the same rate, but if you start with more, you end up
with more. Vice versa, if the sun is particularly active some
decade. Now, this explanation doesn't apply to most kinds of
radioactive dating. Uranium dating, for instance, looks at U-238
that has been there since the beginning of the solar system, and
cosmic rays just don't matter.
