5. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
6. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
e.g. (i) y log3 x 5
7. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
e.g. (i) y log3 x 5
dy 3
dx 3 x 5
8. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
e.g. (i) y log3 x 5 ii y log x 3
dy 3
dx 3 x 5
9. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
e.g. (i) y log3 x 5 ii y log x 3
dy 3 dy 3 x 2
3
dx 3 x 5 dx x
10. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
e.g. (i) y log3 x 5 ii y log x 3
dy 3 dy 3 x 2
3
dx 3 x 5 dx x
3
x
11. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
e.g. (i) y log3 x 5 ii y log x 3 OR y log x 3
2
dy 3 dy 3 x
3
dx 3 x 5 dx x
3
x
12. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
e.g. (i) y log3 x 5 ii y log x 3 OR y log x 3
2
dy
3 dy 3 x
3 y 3 log x
dx 3 x 5 dx x
3
x
13. Differentiating
Logarithms
y log f x y log a f x
dy f x dy f x
dx f x dx log a f x
e.g. (i) y log3 x 5 ii y log x 3 OR y log x 3
2
dy
3 dy 3 x
3 y 3 log x
dx 3 x 5 dx x dy 3
3
dx x
x
16. (iii) y loglog x
1
dy
x
dx log x
1
x log x
17. (iii) y loglog x
1
dy
x
dx log x
1
x log x
iv y log x 3 x 2
18. (iii) y loglog x
1
dy
x
dx log x
1
x log x
iv y log x 3 x 2
dy x 31 x 2 1
dx x 3 x 2
19. (iii) y loglog x
1
dy
x
dx log x
1
x log x
iv y log x 3 x 2
dy x 31 x 2 1
dx x 3 x 2
2x 5
x 3 x 2
20. (iii) y loglog x
1
dy
x
dx log x
1
x log x
iv y log x 3 x 2 OR y log x 3 log x 2
dy x 31 x 2 1
dx x 3 x 2
2x 5
x 3 x 2
21. (iii) y loglog x
1
dy
x
dx log x
1
x log x
iv y log x 3 x 2 OR y log x 3 log x 2
dy x 31 x 2 1 dy 1 1
dx x 3 x 2 dx x 3 x 2
2x 5
x 3 x 2
22. (iii) y loglog x
1
dy
x
dx log x
1
x log x
iv y log x 3 x 2 OR y log x 3 log x 2
dy x 31 x 2 1 dy 1 1
dx x 3 x 2 dx x 3 x 2
2x 5 x 2 x 3
x 3 x 2 x 3 x 2
23. (iii) y loglog x
1
dy
x
dx log x
1
x log x
iv y log x 3 x 2 OR y log x 3 log x 2
dy x 31 x 2 1 dy 1 1
dx x 3 x 2 dx x 3 x 2
2x 5 x 2 x 3
x 3 x 2 x 3 x 2
2x 5
x 3 x 2
25. v y log x 5
x 2
x 21 x 51
dy
x 22
dx x5
x2
26. v y log x 5
x 2
x 21 x 51
dy
x 22
dx x5
x2
3 x 2
x 22 x 5
27. v y log x 5
x 2
x 21 x 51
dy
x 22
dx x5
x2
3 x 2
x 22 x 5
3
x 2 x 5
28. v y log x 5
OR y log x 5 log x 2
x 2
x 21 x 51
dy
x 22
dx x5
x2
3 x 2
x 22 x 5
3
x 2 x 5
29. v y log x 5
OR y log x 5 log x 2
x 2
x 21 x 51 dy
1
1
dy x 22 dx x 5 x 2
dx x5
x2
3 x 2
x 22 x 5
3
x 2 x 5
30. v y log x 5
OR y log x 5 log x 2
x 2
x 21 x 51 dy
1
1
dy x 22 dx x 5 x 2
dx x5 x 2 x 5
x2 x 5 x 2
3 x 2
x 22 x 5
3
x 2 x 5
31. v y log x 5
OR y log x 5 log x 2
x 2
x 21 x 51 dy
1
1
dy x 22 dx x 5 x 2
dx x5 x 2 x 5
x2 x 5 x 2
3 x 2 3
x 22 x 5 x 2 x 5
3
x 2 x 5
32. v y log x 5
OR y log x 5 log x 2
x 2
x 21 x 51 dy
1
1
dy x 22 dx x 5 x 2
dx x5 x 2 x 5
x2 x 5 x 2
3 x 2 3
x 22 x 5 x 2 x 5
3
x 2 x 5
vi y log10 6 x
33. v y log x 5
OR y log x 5 log x 2
x 2
x 21 x 51 dy
1
1
dy x 22 dx x 5 x 2
dx x5 x 2 x 5
x2 x 5 x 2
3 x 2 3
x 22 x 5 x 2 x 5
3
x 2 x 5
vi y log10 6 x
dy 6
dx log 10 6 x
34. v y log x 5
OR y log x 5 log x 2
x 2
x 21 x 51 dy
1
1
dy x 22 dx x 5 x 2
dx x5 x 2 x 5
x2 x 5 x 2
3 x 2 3
x 22 x 5 x 2 x 5
3
x 2 x 5
vi y log10 6 x
dy 6
dx log 10 6 x
1
x log 10
35. v y log x 5
OR y log x 5 log x 2
x 2
x 21 x 51 dy
1
1
dy x 22 dx x 5 x 2
dx x5 x 2 x 5
x2 x 5 x 2
3 x 2 3
x 22 x 5 x 2 x 5
3
x 2 x 5 Exercise 12B; 1acf, 2chk, 5acehi, 6b,
7ad, 8acef, 9bd, 10ac, 11, 13a, 14bdfhjl,
vi y log10 6 x 15b, 18bdf, 19b, 20af*, 21a*
dy 6
dx log 10 6 x Exercise 12C; 1bdf, 2, 3, 6, 7a, 8, 11,
1
13, 14, 18*
x log 10