We're asked to write
each logarithmic expression as a single logarithm.
For number one, we're given log base seven of six,
plus log base seven of eight, minus log base seven of 12.
So the first thing to recognize here,
is that all the logarithms are base seven.
The next thing to recognize is that,
because all the coefficients are one,
or negative one because of the subtraction,
we can go ahead and combine these logarithms
using the product property of logarithms
and the quotient property of logarithms.
Notice here we have a sum of two
logarithms with the same base,
and therefore we can combine them
using the product property of logarithms,
which is from right to left, log base b of m,
plus log base b of n, equals log base b of m times n.
Which means, log base seven of six,
plus log base seven of eight,
is equal to log base seven of six times eight,
we still have minus log base seven of 12.
Six times eight equals 48,
which is equal to log base seven of 48,
minus log base seven of 12.
Now we have a different set of two logarithms,
and therefore we can combine these two logarithms
using the quotient property of logarithms,
which from right to left is log base b of m,
minus log base b of n, equals log base b of m divided by n.
So applying the quotient property to this difference,
this is equal to log base seven of 48 divided by 12,
and 48 divided by 12 is equal to four.
The expression simplifies to log base seven of four.
Let's look at number two on the next slide.
So again, the first thing to recognize about number two,
is that all of the logarithms are base two.
However, all the coefficients are not one,
or negative one, in the case of subtraction,
and therefore to begin, we apply
the power property of logarithms,
which is shown here at the bottom,
and we'll use that property from right to left,
which is n times log base b of m,
equals log base b of m, raised to the power of n.
So for number two, we'll leave the first log
in its current form, which is log base two of two x,
and now for the second log,
because we have a coefficient of 1/2,
we would have plus log base two of the quantity x plus one,
raised to the 1/2 power, and then we have minus,
applying the power property of logarithms here,
we'd have log base two of,
we already have the quantity x plus two squared here,
so we have the quantity x plus two squared,
and this is raised to the second power
because of the two in front.
Here we have powers raised to powers,
which means we multiply the exponents,
and let's write the quantity x plus one to the 1/2
as the square root of the quantity x plus one.
So this is equal to log base two of two x,
plus log base two of the square root
of the quantity x plus one,
and we have minus log base two of the quantity x plus two,
with a power of two times two, which is four.
Now that all coefficients are one,
when the case of the minus negative one,
we can combine these logarithms using
the product property of logarithms,
and the quotient property of logarithms.
So because we have a sum here,
we apply the product property of logarithms,
we can write this as a single log,
by multiplying two x and the square root
of the quantity x plus one.
So this sum is equal to log base two of the quantity
two x times the square root of the quantity x plus one,
we still have minus log base two of the quantity
x plus two, raised to the fourth.
Now we have a difference of two logs,
which means we can combine these two logarithms
using the quotient property of logarithms.
Which means, because we have a difference
of two logarithms with the same base,
we can combine these logarithms by using a quotient.
The difference of these two logs is equal
to log base two of the quantity two x,
times the square root of the quantity x plus one,
divided by the quantity x plus two, raised to the fourth.
We now have the log expression written as a single log.
I hope you found this helpful.