Download 1 a ≠ and 1 log 4 log 0.25

3.3 Properties of Logarithms
Change-of-Base Formula – Let a, b and x be positive real numbers such that a  1 and b  1. Then
log a x can be converted to a different base as follows:
log a x 
logb x log x ln x


logb a log a ln a
Examples: Rewrite the log as a ratio in two ways.
1. log3 47
2. log1/ 3 x
3. log x
3
4
Examples: Evaluate, round to three decimal places.
1. log 7 4
2. log1/ 4 5
3. log 20 0.25
Properties of Logarithms – Let 'a' be a positive number such that a  1 , and let n be a real number. If u
and v are positive real numbers, the following properties are true.
1. Product Property - log a  uv   log a u  log a v or ln  uv   ln u  ln v
2. Quotient Property - log a
u
u
 log a u  log a v or ln  ln u  ln v
v
v
3. Power Property - log a u n  n log a u or ln u n  n ln u
Examples: Use the properties of logarithms to expand the expression.
1. log3 10z
2. log 4x 2 y
3. ln
6
x2  1
Examples: Condense to a single logarithm.
1. log5 8  log5 t
2. log x  2log y  3log z
3. ln x  ln  x  1  ln  x  1
4. You try it:
1
log 4  x  1  2log 4  x  1   6log 4 x
2