Download x x log log =

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5.6 Notes
Laws of Logarithms
PreCalc
Common Logarithm
Natural Logarithm
log is called the common log
*The base is 10
ln is called the natural log
*The base is e
log x  log10 x
ln x  log e x
Solve for each variable knowing the definition of log and ln.
2. ln x  3
1. log 0.1  x
3.
log r  6
4.
log y  e
Properties of Logarithms (just like exponent properties)
log b mn  log b m  log b n
Product:
m
 log b m  log b n
n
Quotient:
log b
Power:
log b m p  p log b m
Property of Equality:
If logb m  logb n
then m  n
log 10  log
log 10 10  1
Special Cases:
log 2 2 3  3
log b 1  0
Making Connections…
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Examples: Expand each logarithm.
1. log 3 5 x
8
2. ln w
3
3. log 2 r
4
4. log x m n
log 5 w x 6
5.
m
ln
6.
n3
a
7. log b 
5
8.
log 2
m
n3
Condense each expression to a single logarithm.
9.
log a 3  log a 4
10.
11.
4 log b 2
12.
ln 6  ln 5  ln 2
13.
ln 7  ln 5
log a 36
2
2 log d w
14.
3
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AFTER you condense, LOOK to see if you can evaluate the logarithm and simplify, if possible.
15. 2 log 3 6  log 3 4
16. 2 log10 5  log10 4
17. log 4 40  log 4 5
18.
Given that log 2 3  1.59
and
log 4 3  log 4 48
log 2 5  2.32 (accurate to two decimal places), find the following.
Steps:
1. Expand the function
2. Evaluate each log
3. Simplify
2
19. log 2 (3 )
21.
log 2 125
5
log
2
20.
3
22.
log 2 15
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5.6 Practice Problems
Expand each logarithm.
1. log 3 m 6 n 3
4. log x
3
2
g h
2. ln ab 
3. log 2 b c
x4
log 5 3
5.
y
l
6. ln jk
4
Condense each expression to a single logarithm. *Don’t forget to evaluate after condensing!
7. log a x  4 log a y
10.
log a 8
3
1
4 ln A  ln B
8.
2
1
9. 3 log b 2  3 log b r
11. 2 log 2 4  log 2 8
If log10 9  0.95 and log10 2  0.30 , find the following:
9
log
10
12.
2
13.
log 10 18
1
log
10
14.
9
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Practice Problem Answers:
1) 6 log 3 m  3 log 3 n
2
1
2)
4 ln a  4 ln b
4 log 5 x  3 log 5 y
4) 3 log x g  3 log x h
5)
x
7) log a y 4
8)
11) 1
12) 0.65
ln A4 B
1
3) log 2 b  2 log 2 c
6)
ln l  ln j  ln k
9) log b
13) 1.25
8
3
r
10) log a 2
14) -0.95
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