# Properties of Logarithms Since logs and exponentials of Properties of Logarithms Since logs and exponentials of the same base are inverse functions of each other they undo each other. f x a Remember that: This means that: inverses undo each each other 2 log 2 5 =5 x f 1 x log a x f f 1 x and f 1 f x ff

1 a log a x x f 1 f log a a x x 7 log 3 3 =7 Properties of Logarithms CONDENSED 1. 2. 3. = EXPANDED log a MN = log a M log a N M log a

N log a M r = log a M log a N = r log a M (these properties are based on rules of exponents since logs = exponents) Using the log properties, write the expression as a sum and/or difference of logs (expand). 4 ab log 6 2 3 c 4 ab log 6 2 3 c

When working with logs, re-write any radicals as rational exponents. 2 using the second property: log a M log a M log a N N using the first property: log a MN log a M log a N log 6 a log 6 b 4 log 6 c using the third property: log a M r r log a M log 6 ab 4 log 6 c 3 2 3 2 log 6 a 4 log 6 b log 6 c 3 Using the log properties, write the expression as a single logarithm (condense).

1 2 log 3 x log 3 y 2 using the third property: log a M r r log a M log 3 x 2 log 3 y this direction using the second property: log a M log a M log a N N this direction log 3 x y 2 1 2 1 2

More Properties of Logarithms This one says if you have an equation, you can take the log of both sides and the equality still holds. If M N , then log a M log a N If log a M log a N , then M N This one says if you have an equation and each side has a log of the same base, you know the "stuff" you are taking the logs of are equal. log 2 8 3 (2 to the what is 8?) There is an answer to this and it must be more than 3 but less than 4, but we can't do this one in our head. Let's put it equal to x and we'll solve for x. log 2 16 4 (2 to the what is 16?) 3.32 log 2 10 (2 to the what is 10?) Check by putting 23.32 in your calculator (we rounded so

it won't be exact) Change to exponential form. log 2 10 x use log property & take log of both sides (we'll use common log) If M N , then log a M log a N use 3rd log property x 2 10 x log 2 log10 log a M r r log a M solve for x by dividing by log 2 use calculator to approximate x log 2 log 10 log 10 3.32

x log 2 If we generalize the process we just did we come up with the: Change-of-Base Formula log b M log a M log b a log M log a ln M ln a The base you change to can common be any base so generally log base 10 well want to change to a LOG base so we can use our calculator. That would be LN either base 10 or base e. natural log base e

Example for TI-83 Use the Change-of-Base Formula and a calculator to approximate the logarithm. Round your answer to three decimal places. log 3 16 Since 32 = 9 and 33 = 27, our answer of what exponent to put on 3 to get it to equal 16 will be something between 2 and 3. ln 16 log 3 16 ln 3 2.524 put in calculator