12 x1 t01 01 log laws (2012)
TRANSCRIPT
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