Fisher, Section 7.4
Algebra 2
7.4 Properties of Logarithms
Name:______________________________
Date:_______________Pd:_____________
Teacher Do
7.4 – Properties of Logarithms
I. Properties of Logarithms (similar to exponent properties)
Product:
log b mn  log b m  log b n
Quotient:
log b
Power:
log b m p  p log b m
m
 log b m  log b n
n
II. Simplifying Logarithms
Write each expression into a single logarithm. Simplify when possible.
coefficients_______________ as exponents inside the log

Rewrite ____

Rewrite addition/subtraction of logs as multiplication/division of bases.
TEACHER DO:
Write as a single logarithm
Example 1: 3log520 - 2log510
log520³- log510²
Step #1: Write coefficient as an exponent
log58000 - log5100
** Remember need to have same base
8000
log5 100
=
Step #2: Rewrite subtraction of log as division of bases
log5 80
1
Fisher, Section 7.4
Example #2: log 2 + log 4 – log 7
log b mn  log b m  log b n
log (2·4) = 8
log 8
log b
7
m
 log b m  log b n
n
8
7
Answer: log
Students Do:
Write as a single logarithm
1. log 7 + log 2
2. log 6 − log 3
log (7·2)= log 14
3.
log 6
log 3
log35=log 243
2
1
= log4
3
6
log4
=2.38
5.
2log26 – log29
6. 6 log 2 𝑥 + 5 log 2 𝑥
=
log262 – log29
=
=
log236– log29
=
= log2 36
= log24
9
2
log42 - log46
4.
5 log 3
= log
log2x6
+
log2x5
log2 (x6 · x5)
= log2 x
11
2
Fisher, Section 7.4
7. log 4 5 + log 4 3
8. log 3 𝑦 + log 3 8𝑥 − log 3 4
log4 (5· 3)
log3 (y· 8x)
log4 15
log3 8xy
4
log3 2xy
III. Expanding Logarithms
 Rewrite multiplication/division of bases as addition and subtraction of logs.
 Bring exponents down in front of the logarithm as a coefficient
Teacher Do
Expand each logarithm
1. log3
12
5
=
log3 12 - log3 5
2. log5 7³=
3log57
3. log4 (4·3)=
log4 4 + log4 3
3
Fisher, Section 7.4
Students Do
Expand each logarithm.
4𝑥
1. log 𝑦
2. log 3
log 4x – log y
log3 250 - log3 37
250
37
3. log 3 9𝑥 5
log b m p  p log b m
5 log39x
IV. Change of Base formula
We can evaluate log 53 » 1.724276 and ln 75 » 4.317488 in our calculator because calculators can evaluate two
types of logarithms, common logarithms (base 10) and natural logarithms (base e). What if we need to evaluate
log3 89 or log7215?
Neither of these problems can simply get typed into the calculator because calculators do not have log base 3 or
log base 7 buttons.
In order to evaluate logarithms with ANY base, we can use the following formula:
log b x 
log a x
log a b
Teacher Do
Evaluate
log389
log389=x
89= 3x
log (89)= log (3x)
log 89=x log (3)
log 89
log 3
=x
4.085
=x
4
Fisher, Section 7.4
STUDENTS DO:
Use the Change of Base formula to evaluate the expression.
1. log 81 27
log8127=x
27=81x
log (27)= log (81x)
log 27=x log (81)
log 27 = x
log 81
x=.75
2. log 5 36
log536=x
36=5x
log (36)= log (5x)
log 36= xlog (5)
log 36
log 5
=x
x=2.22
3. log 2 50
log (50) = log (2x )
log 50= x log 2
log 50
log 2
=x
5
Fisher, Section 7.4
Students-EXTRA Practice- end of class independent work
V. Use properties of logarithms to evaluate each expression.
1. log 2 4 − log 2 16
2. log 6 12 − log 6 2
1
3. log 4 48 − 2 log 4 9
You Try
Write each expression into a single logarithm. Simplify when possible.
1. log 2 9 + log 2 3
2. log 6 12 + log 6 𝑥
3.
log 6 5 − log 6 𝑥
4. log 4 32 − log 4 2
5.
4log 3 2
6. 2 log 7 6 − log 7 9
7. 4 log 𝑚 − 3log 𝑛
8. Log 3 𝑥 + 2log 3 𝑦 − log 3 𝑧
Expand each logarithm.
1. log 7𝑥
𝑟
4. log 5 𝑠
2. log14 𝑥 3 𝑦 5
3. log 7 49𝑥𝑦𝑧
5. log 4 5√𝑥
6.
Use the Change of Base formula to evaluate the expression.
1. log 2 9
2. log12 20
6