12 x1 t01 01 log laws (2012)

Logarithms

LogarithmsLogarithms are the inverse of exponentials.

LogarithmsLogarithms are the inverse of exponentials. If y  a x

LogarithmsLogarithms are the inverse of exponentials. If y  a x then x  log a y

LogarithmsLogarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x

LogarithmsLogarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y

LogarithmsLogarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y

LogarithmsLogarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y x  ln y x  log y

LogarithmsLogarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y log base e is known as x  ln y the natural logarithm. x  log y

LogarithmsLogarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y log base e is known as x  ln y the natural logarithm. x  log y y y  log a x a  1 1 x

LogarithmsLogarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y log base e is known as x  ln y the natural logarithm. x  log y y y  log a x a  1 1 x y  log a x 0  a  1

Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y log base e is known as x  ln y the natural logarithm. x  log y y y  log a x a  1 1 xdomain : x  0 y  log a x 0  a  1

Logarithms Logarithms are the inverse of exponentials. If y  a x then x  log a y If y  e x then x  log e y log base e is known as x  ln y the natural logarithm. x  log y y y  log a x a  1 1 xdomain : x  0range : all real y y  log a x 0  a  1

Log Laws

Log Laws1 log a m  log a n  log a mn

Log Laws1 log a m  log a n  log a mn m2 log a m  log a n  log a   n

Log Laws1 log a m  log a n  log a mn m2 log a m  log a n  log a   n3 log a m n  n log a m

Log Laws1 log a m  log a n  log a mn m2 log a m  log a n  log a   n3 log a m n  n log a m4 log a 1  0

Log Laws1 log a m  log a n  log a mn m2 log a m  log a n  log a   n3 log a m n  n log a m4 log a 1  05 log a a  1

Log Laws1 log a m  log a n  log a mn m2 log a m  log a n  log a   n3 log a m n  n log a m4 log a 1  05 log a a  16 a log x  x a

Log Laws1 log a m  log a n  log a mn m2 log a m  log a n  log a   n3 log a m n  n log a m4 log a 1  05 log a a  16 a log x  x a log b x7  log a x  log b a

e.g. (i) x  log 5 125

e.g. (i) x  log 5 125 5 x  125

e.g. (i) x  log 5 125 5 x  125 x3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16  log 4 4 2

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16  log 4 4 2  2 log 4 4 2

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3  log 4 4 2  2 log 4 4 2

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3  log 4 4 2 6 log 6 32  2 log 4 4 2

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3  log 4 4 2 6 log 6 32  2 log 4 4  32 2

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3  log 4 4 2 6 log 6 32  2 log 4 4  32 2 9

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  2 log 4 4  32 2 9

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32 2 9

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 d) log10125  log10 32  log10 4

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 d) log10125  log10 32  log10 4 125  32   log10     4 

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 d) log10125  log10 32  log10 4 125  32   log10     4   log10 1000

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 d) log10125  log10 32  log10 4 125  32   log10     4   log10 1000 3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 d) log10125  log10 32  log10 4 log 7 8 e) 125  32  log 7 2  log10     4   log10 1000 3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 d) log10125  log10 32  log10 4 log 7 8 e) 125  32  log 7 2  log10     4   log 2 8  log10 1000 3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 d) log10125  log10 32  log10 4 log 7 8 e) 125  32  log 7 2  log10     4   log 2 8  log10 1000 3 3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 1 f) log 2 log 7 8 8 d) log10125  log10 32  log10 4 e) 125  32  log 7 2  log10     4   log 2 8  log10 1000 3 3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 1 f) log 2 log 7 8 8 d) log10125  log10 32  log10 4 e) 1 1 log 7 2  log 2 125  32   log10    2 8  4   log 2 8  log10 1000 3 3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 1 f) log 2 log 7 8 8 d) log10125  log10 32  log10 4 e) 1 1 log 7 2  log 2 125  32   log10    2 8  4   log 2 8 1  log10 1000   3 3 2 3

e.g. (i) x  log 5 125 ii  log x 343  3 5 x  125 x 3  343 x3 x7iii  Evaluate; a) log 4 16 b) 6 2log 6 3 c) log 216  log 2 8  log 4 4 2 6 log 6 32  log 2 128  2 log 4 4  32  log 2 27 2 9 7 1 f) log 2 log 7 8 8 d) log10125  log10 32  log10 4 e) 1 1 log 7 2  log 2 125  32   log10    2 8  4   log 2 8 1  log10 1000   3 3 2 3 3  2

1iv  32 x 1  27

1iv  32 x 1  27 32 x 1  33

1iv  32 x 1  27 32 x 1  33 2 x  1  3 2 x  4 x  2

1iv  32 x 1  v  2 x  9 27 32 x 1  33 2 x  1  3 2 x  4 x  2

1iv  32 x 1  v  2 x  9 27 log 2 x  log 9 32 x 1  33 2 x  1  3 2 x  4 x  2

1iv  32 x 1  v  2 x  9 27 log 2 x  log 9 32 x 1  33 x log 2  log 9 2 x  1  3 2 x  4 x  2

1iv  32 x 1  v  2 x  9 27 log 2 x  log 9 32 x 1  33 x log 2  log 9 2 x  1  3 log 9 2 x  4 x log 2 x  2

1iv  32 x 1  v  2 x  9 27 log 2 x  log 9 32 x 1  33 x log 2  log 9 2 x  1  3 log 9 2 x  4 x log 2 x  2 x  3.17 (to 2 dp)

1iv  32 x 1  v  2 x  9 27 log 2 x  log 9 32 x 1  33 x log 2  log 9 2 x  1  3 log 9 2 x  4 x log 2 x  2 x  3.17 (to 2 dp) Exercise 12A; 2, 3aceg, 4bdfh, 5ab, 6ab, 7ac, 8bdh, 9ac, 14, 18* Exercise 6B; 8

12 x1 t01 01 log laws (2012)

by Nigel Simmons on Feb 03, 2012

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12 x1 t01 01 log laws (2012) — Webinar Transcript