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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.

**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 **

a) log 3 Use the laws of logarithms to simplify and then evaluate: 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

**Example **

Write the given expression in terms of log a x , log a y , 5 and log a z : log a 5 x 2 y 4 z

**CONVERSION TO BASE 10:**

Proof: log a x log x log a Ex. log 2 5

** Homework: **

p.475–476 #1 – 8, 11