12x1 t01 01 log laws

Logarithms

LogarithmsLogarithms are the inverse of exponentials.

Logarithmsxay If

Logarithms are the inverse of exponentials.

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If yx elogthen

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If yx elogthen

yx ln

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If yx elogthen

yx lnyx log

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If yx elogthen

yx lnyx log

log base e is known as the natural logarithm.

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If yx elogthen

yx lnyx log

log base e is known as the natural logarithm.

y

x1

1 log axy a

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If yx elogthen

yx lnyx log

log base e is known as the natural logarithm.

y

x1

1 log axy a

10 log axy a

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If yx elogthen

yx lnyx log

log base e is known as the natural logarithm.

y

x1

1 log axy a

10 log axy a

0:domain x

Logarithmsxay If yx alogthen

Logarithms are the inverse of exponentials.

xey If yx elogthen

yx lnyx log

log base e is known as the natural logarithm.

y

x1

1 log axy a

10 log axy a

0:domain xy realall:range

Log Laws

mnnm aaa logloglog 1

Log Laws

mnnm aaa logloglog 1

Log Laws

nmnm aaa logloglog 2

mnnm aaa logloglog 1

Log Laws

nmnm aaa logloglog 2

mnm an

a loglog 3

mnnm aaa logloglog 1

Log Laws

nmnm aaa logloglog 2

mnm an

a loglog 3

01log 4 a

mnnm aaa logloglog 1

Log Laws

nmnm aaa logloglog 2

mnm an

a loglog 3

01log 4 a

1log 5 aa

mnnm aaa logloglog 1

Log Laws

nmnm aaa logloglog 2

mnm an

a loglog 3

01log 4 a

1log 5 aa

xa xa log 6

mnnm aaa logloglog 1

Log Laws

nmnm aaa logloglog 2

mnm an

a loglog 3

01log 4 a

1log 5 aa

xa xa log 6

axx

b

ba log

loglog 7

e.g. (i) 125log5x

e.g. (i) 125log5x1255 x

e.g. (i) 125log5x1255 x

3x

e.g. (i) 125log5x1255 x

3x

3343log xii

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 4

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log6

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log10

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

2log8log e)

7

7

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

2log8log e)

7

7

8log2

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

2log8log e)

7

7

8log23

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

2log8log e)

7

7

8log23

81log f) 2

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

2log8log e)

7

7

8log23

81log f) 2

81log

21

2

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

2log8log e)

7

7

8log23

81log f) 2

81log

21

2

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

2log8log e)

7

7

8log23

81log f) 2

81log

21

2

321

e.g. (i) 125log5x1255 x

3x

3343log xii3433 x7x

Evaluate; iii16log a) 4

24 4log

4log2 42

32log66 b)2

6 3log623

9

8log16log c) 22 128log2

72 2log

7

4log32log125log d) 101010

4

32125log10

1000log103

2log8log e)

7

7

8log23

81log f) 2

81log

21

2

321

23

2713 12 xiv

312 33 x

2713 12 xiv

312 33 x

2713 12 xiv

242312

xx

x

312 33 x

2713 12 xiv

242312

xx

x

92 xv

312 33 x

2713 12 xiv

242312

xx

x

92 xv

9log2log x

312 33 x

2713 12 xiv

242312

xx

x

92 xv

9log2log x

9log2log x

312 33 x

2713 12 xiv

242312

xx

x

92 xv

9log2log x

9log2log x

2log9log

x

312 33 x

2713 12 xiv

242312

xx

x

92 xv

9log2log x

9log2log x

2log9log

x

dp)2(to 17.3x

312 33 x

Exercise 12A; 2, 3aceg, 4bdfh, 5ab, 6ab, 7ac, 8bdh, 9ac, 14, 18*

Exercise 6B; 8

2713 12 xiv

242312

xx

x

92 xv

9log2log x

9log2log x

2log9log

x

dp)2(to 17.3x