# The LOGARITHMIC LAWS

### 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

 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