The LOGARITHMIC LAWS
Since the logarithm function is the inverse of the exponential function, each exponent law has a corresponding logarithmic law . Logarithmic expressions must have a common base when they are simplified or evaluated using the laws of logarithms.
LAWS of LOGARITHMS
1. PRODUCT LAW:
Proof: log a xy  log a x  log a y Ex.
log
2
( 5 x 3 )
2. QUOTIENT LAW: log a

 x y


 log a x  log a y Ex.
log
4
8
5
3. POWER LAW: log a x n
 n log a x Ex.
log
3
5
The laws of logarithms can be used both forward and backward to simplify and evaluate expressions.
Example  Evaluate each of the following: a) log
5
3
25 b) log
3
9 27

Example  Use the laws of logarithms to simplify and then evaluate: a) log
3
54  log
3
2 b) log 25  log 8  log 2 c) log
2
20  log
2
5
4 d) 3 log
4
2  log
4
10  log
4
5
Example  Write the given expression in terms of log a x , log a y , and log a z : log a
5 x
2 y
4 z
CONVERSION TO BASE 10: x  log x log a
Proof: log a
Ex. log
2
5
Homework: p.475–476 #1 – 8, 11