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Math 170 - Cooley Pre-Calculus OCC Section 4.3 – Rules of Logarithms Inverse Rules If a 0 and a 1 , then 1. loga (a) x x for any real number a . 2. aloga ( x ) x for x 0 . Product Rule For Logarithms For M 0 and N 0 , log a ( MN ) log a M log a N . (The logarithm of a product is the sum of the logarithms of the factors.) Quotient Rule For Logarithms For M 0 and N 0 , M log a N log a M log a N . (The logarithm of a quotient is logarithm of the dividend minus the logarithm of the divisor.) Power Rule For Logarithms For M 0 any real number N, log a M N N log a M . (The logarithm of a power of M is the exponent times the logarithm of M.) Rules of Logarithms with Base a If M, N, and a are positive real numbers with a 1 , and x is any real number, then, 1. log a (a) 1 5. log a ( MN ) log a M log a N 2. log a (1) 0 M 6. log a N 3. log a ( a ) x x 7. log a M x x log a M 4. aloga ( N ) N 1 8. log a log a N N log a M log a N -1- Math 170 - Cooley Pre-Calculus OCC Section 4.3 – Rules of Logarithms Rules of Natural Logarithms If M, and N are positive real numbers and x is any real number, then, 1. ln(e) 1 5. ln(e x ) x ln( MN ) ln M ln N 2. ln(1) 0 M 6. ln N 3. ln(e x ) x 7. ln M x x ln M 4. eln( N ) N 1 8. ln ln N N Base-Change Formula If a 0, b 0, a 1, b 1 and M 0 , then log a M (To find the log, base b, of M, we typically compute ln M ln N logb M logb a . ln M log M or .) ln a log a Exercises: Simplify each expression. 1) 10log(3x1) 2) log 4 2300 4) log( x 2 4) log( x 2) Rewrite each expression as a single logarithm. 3) ln(6) ln(2) Rewrite each expression as a sum or difference of logarithms. 5) log3 xy 6) ab ln b -2- Math 170 - Cooley Pre-Calculus OCC Section 4.3 – Rules of Logarithms Exercises: Rewrite each expression as a sum or difference of multiple logarithms. 7) log 2 xyz 8) 4a log 3b 9) 5 x 2 ln 2 10) 3x y log 4 3 x 1 Rewrite each expression as a single logarithm. 11) 3log5 p 12 log5 t log5 7 12) 1 log 2 x 3(log a a 2 x loga y) 13) ln(2) ln(3) ln(5) ln(7) 14) 1 log( x) log( y) log z 2 Find an approximate rational solution to equation. Round answers to four decimal places. 15) 3x 12 16) 1.09x 3 -3- Math 170 - Cooley Pre-Calculus OCC Section 4.3 – Rules of Logarithms Exercises: Use a calculator and the change-of-base formula to find each logarithm to four decimal places. 17) log 3 (4.78) 18) log1.2 (13.7) Write each equation as an equivalent logarithmic equation. 19) 21) (1.025)12t 3 20) e x 456 Find the time it takes $10,000 to grow to $10,000,000 with a rate of 7% that is compounded quarterly. -4-