Properties of Logarithms Since logs and exponentials of

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

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