12 x1 t01 01 log laws (2013)
TRANSCRIPT
![Page 1: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/1.jpg)
Logarithms
![Page 2: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/2.jpg)
LogarithmsLogarithms are the inverse of exponentials.
![Page 3: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/3.jpg)
Logarithmsxay If
Logarithms are the inverse of exponentials.
![Page 4: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/4.jpg)
Logarithmsxay If yx alogthen
Logarithms are the inverse of exponentials.
![Page 5: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/5.jpg)
Logarithmsxay If yx alogthen
Logarithms are the inverse of exponentials.
xey If
![Page 6: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/6.jpg)
Logarithmsxay If yx alogthen
Logarithms are the inverse of exponentials.
xey If yx elogthen
![Page 7: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/7.jpg)
Logarithmsxay If yx alogthen
Logarithms are the inverse of exponentials.
xey If yx elogthen
yx ln
![Page 8: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/8.jpg)
Logarithmsxay If yx alogthen
Logarithms are the inverse of exponentials.
xey If yx elogthen
yx lnyx log
![Page 9: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/9.jpg)
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.
![Page 10: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/10.jpg)
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
![Page 11: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/11.jpg)
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
![Page 12: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/12.jpg)
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
![Page 13: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/13.jpg)
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
![Page 14: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/14.jpg)
Log Laws
![Page 15: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/15.jpg)
mnnm aaa logloglog 1
Log Laws
![Page 16: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/16.jpg)
mnnm aaa logloglog 1
Log Laws
nmnm aaa logloglog 2
![Page 17: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/17.jpg)
mnnm aaa logloglog 1
Log Laws
nmnm aaa logloglog 2
mnm an
a loglog 3
![Page 18: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/18.jpg)
mnnm aaa logloglog 1
Log Laws
nmnm aaa logloglog 2
mnm an
a loglog 3
01log 4 a
![Page 19: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/19.jpg)
mnnm aaa logloglog 1
Log Laws
nmnm aaa logloglog 2
mnm an
a loglog 3
01log 4 a
1log 5 aa
![Page 20: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/20.jpg)
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
![Page 21: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/21.jpg)
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
![Page 22: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/22.jpg)
e.g. (i) 125log5x
![Page 23: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/23.jpg)
e.g. (i) 125log5x1255 x
![Page 24: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/24.jpg)
e.g. (i) 125log5x1255 x
3x
![Page 25: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/25.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii
![Page 26: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/26.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x
![Page 27: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/27.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
![Page 28: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/28.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
Evaluate; iii16log a) 4
![Page 29: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/29.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
Evaluate; iii16log a) 4
24 4log
![Page 30: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/30.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
Evaluate; iii16log a) 4
24 4log
4log2 4
![Page 31: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/31.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
Evaluate; iii16log a) 4
24 4log
4log2 42
![Page 32: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/32.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
Evaluate; iii16log a) 4
24 4log
4log2 42
32log66 b)
![Page 33: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/33.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
Evaluate; iii16log a) 4
24 4log
4log2 42
32log66 b)2
6 3log6
![Page 34: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/34.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
Evaluate; iii16log a) 4
24 4log
4log2 42
32log66 b)2
6 3log623
![Page 35: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/35.jpg)
e.g. (i) 125log5x1255 x
3x
3343log xii3433 x7x
Evaluate; iii16log a) 4
24 4log
4log2 42
32log66 b)2
6 3log623
9
![Page 36: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/36.jpg)
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
![Page 37: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/37.jpg)
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
![Page 38: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/38.jpg)
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
![Page 39: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/39.jpg)
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
![Page 40: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/40.jpg)
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
![Page 41: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/41.jpg)
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
![Page 42: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/42.jpg)
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
![Page 43: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/43.jpg)
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
![Page 44: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/44.jpg)
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
![Page 45: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/45.jpg)
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
![Page 46: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/46.jpg)
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
![Page 47: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/47.jpg)
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
![Page 48: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/48.jpg)
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
![Page 49: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/49.jpg)
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
![Page 50: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/50.jpg)
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
![Page 51: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/51.jpg)
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
![Page 52: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/52.jpg)
2713 12 xiv
![Page 53: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/53.jpg)
312 33 x
2713 12 xiv
![Page 54: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/54.jpg)
312 33 x
2713 12 xiv
242312
xx
x
![Page 55: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/55.jpg)
312 33 x
2713 12 xiv
242312
xx
x
92 xv
![Page 56: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/56.jpg)
312 33 x
2713 12 xiv
242312
xx
x
92 xv
9log2log x
![Page 57: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/57.jpg)
312 33 x
2713 12 xiv
242312
xx
x
92 xv
9log2log x
9log2log x
![Page 58: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/58.jpg)
312 33 x
2713 12 xiv
242312
xx
x
92 xv
9log2log x
9log2log x
2log9log
x
![Page 59: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/59.jpg)
312 33 x
2713 12 xiv
242312
xx
x
92 xv
9log2log x
9log2log x
2log9log
x
dp)2(to 17.3x
![Page 60: 12 x1 t01 01 log laws (2013)](https://reader035.vdocuments.us/reader035/viewer/2022081401/558fc7bf1a28ab6f4e8b477a/html5/thumbnails/60.jpg)
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