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Math 170 - Cooley
Pre-Calculus
OCC
Section 4.3 – Rules of Logarithms
Inverse Rules
If a  0 and a  1 , then
1. loga (a) x  x for any real number a .
2. aloga ( x )  x for x  0 .
Product Rule For Logarithms
For M  0 and N  0 ,
log a ( MN )  log a  M   log a  N  .
(The logarithm of a product is the sum of the logarithms of the factors.)
Quotient Rule For Logarithms
For M  0 and N  0 ,
M
log a 
N

  log a  M   log a  N  .

(The logarithm of a quotient is logarithm of the dividend minus the logarithm of the divisor.)
Power Rule For Logarithms
For M  0 any real number N,
log a  M N   N  log a  M  .
(The logarithm of a power of M is the exponent times the logarithm of M.)
Rules of Logarithms with Base a
If M, N, and a are positive real numbers with a  1 , and x is any real number, then,
1. log a (a)  1
5. log a ( MN )  log a  M   log a  N 
2. log a (1)  0
M
6. log a 
N
3. log a ( a ) x  x
7. log a  M x   x  log a  M 
4. aloga ( N )  N
1
8. log a     log a  N 
N

  log a  M   log a  N 

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Math 170 - Cooley
Pre-Calculus
OCC
Section 4.3 – Rules of Logarithms
Rules of Natural Logarithms
If M, and N are positive real numbers and x is any real number, then,
1. ln(e)  1
5. ln(e x )  x ln( MN )  ln  M   ln  N 
2. ln(1)  0
M
6. ln 
N
3. ln(e x )  x
7. ln  M x   x  ln  M 
4. eln( N )  N
1
8. ln     ln  N 
N
Base-Change Formula
If a  0, b  0, a  1, b  1 and M  0 , then
log a  M  
(To find the log, base b, of M, we typically compute

  ln  M   ln  N 

logb  M 
logb  a 
.
ln M
log M
or
.)
ln a
log a
 Exercises:
Simplify each expression.
1)
10log(3x1)
 
2)
log 4 2300
4)
log( x 2  4)  log( x  2)
Rewrite each expression as a single logarithm.
3)
ln(6)  ln(2)
Rewrite each expression as a sum or difference of logarithms.
5)
log3  xy 
6)
 ab
ln 

 b 
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Math 170 - Cooley
Pre-Calculus
OCC
Section 4.3 – Rules of Logarithms
 Exercises:
Rewrite each expression as a sum or difference of multiple logarithms.
7)
log 2  xyz 
8)
 4a 
log  
 3b 
9)
 5 x 2 
ln 
 2 


10)
 3x y 
log 4  3

 x 1 


Rewrite each expression as a single logarithm.
11)
3log5 p  12 log5 t  log5 7
12)
1 log 2 x  3(log
a
a
2
x  loga y)
13)
ln(2)  ln(3)  ln(5)  ln(7)
14)
1
log( x)  log( y)  log z
2
Find an approximate rational solution to equation. Round answers to four decimal places.
15)
3x  12
16)
1.09x  3
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Math 170 - Cooley
Pre-Calculus
OCC
Section 4.3 – Rules of Logarithms
 Exercises:
Use a calculator and the change-of-base formula to find each logarithm to four decimal places.
17)
log 3 (4.78)
18)
log1.2 (13.7)
Write each equation as an equivalent logarithmic equation.
19)
21)
(1.025)12t  3
20)
e x  456
Find the time it takes $10,000 to grow to $10,000,000 with a rate of 7% that is compounded quarterly.
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