© 2008 Shirley Radai

Properties of Logarithms

What is a logarithm?

Logarithms are really powers (exponents). The Relationship: "logb(x) = y" means the same thing as "by = x".

Since 23=8, 3 is called the logarithm of 8 with base 2.  We write 3=log28.

© 2008 Shirley Radai

Expanding Logarithmic Expressions

To “expand” a logarithmic expression means to take a log with multiple factors inside it and rewrite it into different logs with

single factors inside.

log ( ) log ( ) log ( )mn m nb b b Multiplying inside a log turns into addition outside the log if the bases are the same.

log log ( ) log ( )m

nm nb b b

Division inside a log turns into subtraction outside the log if the bases are the same.

log ( ) log ( )nm n mb bAn exponent inside a log is moved to the front of the log to become a multiplier if the bases are the same.

© 2008 Shirley Radai

Examples

log (5 )3log 5 log3 3

x

x

Since we have multiplication inside the log (5x), it becomes addition.

16log4

log 16 log4 42 log4

xx

x

Since we have division inside the log (16/x), it becomes subtraction.

© 2008 Shirley Radai

Examples (cont’d)

4log ( )64log6

x

x

Since there is an exponent inside x4, the exponent goes out front of the log.

© 2008 Shirley Radai

Properties of Logarithms

logb(b) = 1, for any base b, because b1 = b.

logb(1) = 0, for any base b, because b0 = 1.

logb(a) is undefined if a is negative.

logb(0) is undefined for any base b.

logb(bn) = n, for any base b.

© 2008 Shirley Radai