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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