2. Logarithms of Products
The first property we discuss is related to the product rule for exponents:
Lets examine
log3(9 · 27) vs
log39 + log327.
Note that
log3(9 · 27) = log3243 = 5
35 = 243
and that
log39 + log327 = = 5.
32 = 9 and 33 = 27 So
log3(9 · 27) =
log39 + log327.

3. Example
that is a single logarithm:
Express as an equivalent expression
loga6 + loga7.
Solution
Using the product rule for logarithms
loga6 + loga7 = loga(6 · 7)
= loga(42).

4. Logarithms of Powers
The second basic property is related to the power rule for exponents:

5. Example
expression that is a product:
Use the power rule to write an equivalent
a) loga6–3;
Solution
Using the power rule for logarithms
a) loga6-3 = –3loga6
= log4x1/2
Using the power rule for logarithms
= ½ log4x

6. Logarithms of Quotients
The third property that we study is similar to the quotient rule for exponents:

8. Example
Express as an equivalent expression that is a difference of logarithms:
log3(9/y).
Solution
Using the quotient rule for logarithms
log3(9/y)
= log39 – log3y.
Example
Express as an equivalent expression that is a single logarithm:
loga6 – loga7.
Solution
Using the quotient rule for logarithms “in reverse”
loga6 – loga7 = loga(6/7)

9. Using the Properties Together
Example
using individual logarithms of x, y, and z.
Expand to an equivalent expression
Solution
= log4x3 – log4 yz
= 3log4x – log4 yz
= 3log4x – (log4 y + log4z)
= 3log4x – log4 y – log4z

10. Solution continued

11. Example
using a single logarithm.
Condense to an equivalent expression
Solution
= logbx1/3 – logb y2 + logbz

12. Example
Simplify: a) log b) log33–3.4
Solution
8 is the exponent to which you raise 6 in order to get 68.
a) log668 = 8
b) log33–3.4 = –3.4