In this video, we will look at expanding and
condensing logarithms. First off, here are
the properties written out.
Number 1: The Product Rule for Logarithms.
The log, base b, of M times N is equal to
the log, base b, of M plus the log, base b,
of N.
Number 2: The Quotient Rule for Logarithms.
The log, base b, of M divided by N is equal
to the log, base b, of M minus the log, base
b, of N.
Number 3: The Power Rule for Logarithms. The
log, base b, of M to the power of p is equal
to p times the log, base b, of M.
Now, for a few examples of expanding logarithms:
First, let us look at the log, base b, of
x-squared times y-cubed. By the Product Rule,
this makes the log, base b, of x-squared plus
the log, base b, of y-cubed. By the Power
Rule, applied to the x-squared and y-cubed,
will yield two times the log, base b, of x
plus three times the log, base b, of y.
Secondly, let us look at the log, base b,
of two times the square root of x divided
by y. By the Product Rule, this makes the
log, base b, of two plus the log, base b,
of the square root of x divided by y. Now,
by the Power Rule, this makes the log, base
b, of two plus one-half times the log, base
b, of x divided by y. Finally, by the Quotient
Rule, we have the log, base b, of two plus
one-half times the difference of the log,
base b, of x and the log, base b, of y.
Now, for a few examples of condensing logarithms:
First, let us look at the log, base b, of
x plus two times the log, base b, of z. By
the Power Rule, we have the log, base b, of
x plus the log, base b, of z-squared. Then,
by the Product Rule, we have the log, base
b, of x times z-squared. Secondly, let us
look at three times the log, base b, of x
minus a half times the log, base b, of z.
By the Power Rule, applied twice, we have
the log, base b, of x-cubed minus the log,
base b, of the square-root of z. By the Quotient
Rule, we have the log, base b, of x-cubed
divided by the square-root of z.
