Transcript Slide 1

Properties of Logarithms
© 2008 Shirley Radai
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 (mn)  log (m)  log (n)
b
b
b
Multiplying inside a log turns into addition
outside the log if the bases are the same.
log    log (m)  log (n)
b n 
b
b
Division inside a log turns into subtraction
outside the log if the bases are the same.
m
log (m )  n log (m)
b
b
n
© 2008 Shirley Radai
An exponent inside a log is moved to the
front of the log to become a multiplier if the
bases are the same.
Examples
log (5x)
3
log 5  log x
3
3
 16 
log  
4 x 
log 16  log x
4
4
2  log x
4
© 2008 Shirley Radai
Since we have multiplication inside
the log (5x), it becomes addition.
Since we have division inside the
log (16/x), it becomes subtraction.
Examples (cont’d)
4
log ( x )
6
4log x
6
© 2008 Shirley Radai
Since there is an exponent
inside x4, the exponent goes out
front of the log.
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