logarithms of products the first property we discuss is related to the product rule for exponents:...
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Logarithms of Products
The first property we discuss is related to the product rule for exponents:
.m n m nb b b
Lets examine
log3(9 · 27) vs log39 + log327.
Note that
log3(9 · 27) = log3243 = 5 35 = 243
and that
log39 + log327 = 2 + 3 = 5. 32 = 9 and 33 = 27
log3(9 · 27) = log39 + log327.
So
Solution
= loga(42).
Using the product rule for logarithms
Example
that is a single logarithm:
Express as an equivalent expression
loga6 + loga7.
loga6 + loga7 = loga(6 · 7)
Logarithms of Powers
The second basic property is related to the power rule for exponents:
.nm mnb b
Solution
= log4x1/2
Using the power rule for logarithms
a) loga6-3 = –3loga6
Example
expression that is a product:
Use the power rule to write an equivalent
a) loga6–3;
4b) log .x
4b) log x
= ½ log4x Using the power rule
for logarithms
Logarithms of Quotients
The third property that we study is similar to the quotient rule for exponents:
.m
m nn
bb
b
Solution
log3(9/y)
Example
Express as an equivalent expression that is a difference of logarithms:
log3(9/y).
= log39 – log3y. Using the quotient
rule for logarithms
Solution
Example
Express as an equivalent expression that is a single logarithm:
loga6 – loga7.
loga6 – loga7 = loga(6/7) Using the quotient
rule for logarithms “in reverse”
Solution
= log4x3 – log4 yz
Example
using individual logarithms of x, y, and z.
Expand to an equivalent expression
334 7
a) log b) logbx xy
yz z
3
4a) log x
yz
= 3log4x – log4 yz
= 3log4x – (log4 y + log4z)
= 3log4x – log4 y – log4z
Using the Properties Together
1/ 33
7 7 b) log logb b
xy xy
z z
71
log3 b
xy
z
71log log
3 b bxy z
1log log 7log
3 b b bx y z
Solution continued
1 1 7log log log
3 3 3b b bx y z
Solution
Example
using a single logarithm.
Condense to an equivalent expression
1log 2log log
3 b b bx y z
1log 2log log
3 b b bx y z = logbx1/3 – logb y2 + logbz
1/ 3
2log logb b
xz
y
3
2logb
z x
y
Solution
Example Simplify: a) log668 b) log33–3.4
a) log668 = 8
b) log33–3.4 = –3.4
8 is the exponent to which you raise 6 in order to get 68.