Directions. Expand each logarithm.
Expanding a logarithm means splitting one logarithm into multiple logarithms. It means breaking up a
more complicated operation (like multiplication or division) into several less complicated operations
(like addition or subtraction).
1. log9(a ∙ b ∙ c2)
6
2. log8
Use the Product Property to break up this
logarithm. Instead of the log of 3 things multiplied
together, you will end up with 3 logs being added:
log9a + log9b + log9c2
Use the Power Property to turn the exponent on
everything into a multiplier. (Bring 6 to the front)
Next use the Power Property to turn the exponent
on c into a multiplier. (Bring 2 to the front)
log9a + log9b + 2log9c
Next use the Quotient Property to break up this
logarithm. Instead of two things being divided, you
will end up with 2 logs being subtracted (the
numerator minus the denominator). Leave the 6 in
front of both logs.
6log8x4 – 6log8y
6log8
Finally use the Power Property to turn the
exponent on x into a multiplier. (Bring the 4 out
front of the log – with the 6). Multiply 6 ∙ 4 = 24.
24log8x – 6log8y
3. log9(w ∙
4. log6(z4
First, you should recall from Unit 5 that a radical
with an index of 3 (a.k.a. a “cube root”) is another
way of writing the exponent . That means
is
Just like in Problem #3, you should recall from Unit
5 that a radical with an index of 2 (a.k.a. a “square
root”) is another way of writing the exponent .
the same as
That means
log9(w ∙
.
)
Use the Product Property to break up this
logarithm into three logs added together. Use the
Power Property at the same time to bring
exponents to the front of the logs for u and v which
both have the exponent .
log9w + log9u + log9v
Directions. Condense each logarithm.
log6(z4
)
is the same as
.
)
Use the Product Property to break up this
logarithm into two logs added together. Use the
Power Property at the same time to bring
exponents to the front of the logs for z and x.
4log6z + log6x
Condensing a logarithm means rewriting multiple logarithms (that all have the same base) as just one
logarithm. Simple logs that are added together will be rewritten as the log of a multiplication problem.
Subtraction of logs will be rewritten as the log of a division problem.
5. 3log6u – 3log6v
6. log4c + log4a + log4b
First use the reverse of the Power Property to make First use the reverse of the Power Property to make
the coefficients into exponents.
the coefficients into exponents.
log6u3 – log6v3
log4c + log4 + log4
Next use the reverse of the Quotient Property to
make this into one logarithm.
Next, use the revere Product Property to make this
log6
into one logarithm.
Since both the numerator and denominator have
the same exponent, this can be rewritten as:
log6
log4c
Since a and b both have the same exponent, this can
be rewritten as:
log4
Finally, recall from Unit 5 that a rational (fraction)
exponent like can be rewritten as a square root.
7. 4log3x + 2log3x – 5log3y – log3x
log4c
8. 6log3 – 4log3 + 2log5
First use the reverse of the Power Property to make First use the reverse of the Power Property to make
the coefficients into exponents.
the coefficients into exponents.
log3x4 + log3x2 – log3y5 – log3x
log36 – log34 + log52
Next use the reverse Product Property and the
reverse of the Quotient Property to make one
logarithm with all the positives in the numerator
and all of the negatives in the denominator.
Next use the reverse Product Property and the
reverse of the Quotient Property to make one
logarithm with all the positives in the numerator
and all of the negatives in the denominator.
log
Use the properties of exponents to add the
exponents on top for the x’s to get x6.
Use the properties of exponents to subtract the
exponents from top to bottom for the x’s to get x5.
Use the properties of exponents to subtract the
exponents from top to bottom for the 3’s to get 32.
log3252
Simplify 32 = 9 and 52 = 25. 9 ∙ 25 = 225.
log225
Finally, since the top and bottom have the same
exponent, this can be rewritten as:
Directions. Use the change of base formula to evaluate each expression. Round answers to the nearest hundredth.
Use the formula to enter these logarithms on the calculator and get an approximate decimal answer.
9. log424
10. log3207
On your calculator type: log24
log4
2.29
On your calculator type: log207
log3
4.85