This is a demonstration of several examples of using log rules to handle logs mechanically.
There are two primary (mechanical) uses of logs. The most common usage is to convert a large product of things into a sum of things, but this usage primarily occurs in calculus. For our course, the primary usage will be to expand, simplify, and/or condense logs. This is best demonstrated with some examples.
Note that , and we will use the log rules to remove powers, to get;
Finally, we can combine like terms.
As before, we will begin by expanding each of the logs fully. Also like before,
we must fully factor the argument of the log before we can apply the log
rules to expand it. For the sake of clarity I will factor and then simplify
the first log on it’s own, and then do the same with the second log on it’s
Next we do the same with the second log;
Finally we combine the two expanded expressions we got above;
In order to condense separated log terms we need to get them into the exactly correct
forms for the log rules, that is, either stuffstuff or stuffstuff. In particular, we need to
make sure there are no coefficients in front of the log term. To accomplish this, we
will use the fact that, for any coefficient we can transform into , ie we can move the
coefficient to the argument as a power. Thus, for our expression we would start with,
Note that we could have left the negative sign and written instead of . There is
nothing wrong with doing this, but I find that often the negative sign generates
computational errors, so putting it into the power is a way to avoid that
Next we will condense the logs using the property that in ‘reverse’, meaning we will go from the right hand side to the left hand side of the equality, which gives us; Now we are done. is the condensed form of the log.