Differential Calculus 201-103-RE
Vincent Carrier
Exercise Sheet 19
5.6 Exponential and Logarithmic Equations
Solve the following equations.
1. 2e−3x + 1 = 5
2. (ex − 3)(ex − 5) = 0
3. (ex − 2)(ex + 4) = 0
4. ee − π = 0
5. e2x − 5ex + 6 = 0
6. 3 ln 2x + 7 = −2
7. 4 ln(−3x) − 3 = 5
8. ln ln x − e = 0
9. ln eln e
x
ln x
=3
Find the value(s) of x at which the tangent line is horizontal.
10. y = e5x − e3x
13. y =
ln3 2x
x4
11. y = x3 ln2 4x
14. y =
12. y = ln e6x − 2x
e4x
e2x − 3
15. y = 2 ln x + 5 ln ln(3x)
16. Find the equation of the tangent line to the curve
y = e−2x
parallel to the line
y = −6x − 9.
17. Find the equation of the tangent line to the curve
y = x ln 5x
perpendicular to the line
1
y = x + 7.
2
5.7 Logarithmic Differentiation
Find the derivative of the following functions.
18. y = xx
2
21. y = (x2 + 1)
19. y = (ln x)e
√
x
x
22. y = (arctan x)tan x
x
24. y = xx ln x
25. y = xx
r
27. y = x2 e3x 45x 67x
28. y =
3
sec2 x tan4 x
ex arcsin x
20. y =
√ 1/x
x
23. y = (cos 3x)sin 2x
26. y = ex
29. y =
ex
x5 e−4x ln3 2x
sin2 3x cos4 5x
Answers:
ln 2
3
2. x = ln 3, ln 5
3. x = ln 2
4. x = ln ln π
5. x = ln 2, ln 3
6. x =
1. x = −
7. x = −
e2
3
8. x = ee
dy
1
10.
= e3x (5e2x − 3); x = ln
dx
2
3
5
e
1
2e3
9. x = e3
11.
dy
1
1
= x2 ln 4x (3 ln 4x + 2); x = 0, , 2/3
dx
4 4e
12.
dy
2(3e6x − 1)
ln 3
= 6x
; x=−
dx
e − 2x
6
13.
1 e3/4
dy
ln2 2x (3 − 4 ln 2x)
; x= ,
=
5
dx
x
2 2
14.
dy
2e4x (e2x − 6)
ln 6
=
; x=
dx
(e2x − 3)2
2
15.
dy
2 ln 3x + 5
1
=
; x = 5/2
dx
x ln 3x
3e
16. y = −6x + 3(1 − ln 3)
18.
2
dy
= xx x(2 ln x + 1)
dx
17. y = −2x −
19.
1
5e3
x
dy
1
= (ln x)e ex ln ln x +
dx
x ln x
2
√
1 − ln x
dy
2x3/2
2
x ln(x + 1)
√
= (x + 1)
21.
+ 2
2x2
dx
x +1
2 x
dy
tan x
tan x
2
22.
= (arctan x)
sec x ln arctan x +
dx
(1 + x2 ) arctan x
√ 1/x
dy
=
x
20.
dx
23.
dy
= (cos 3x)sin 2x (2 cos 2x ln cos 3x − 3 sin 2x tan 3x)
dx
dy
= xx ln x ln x (ln x + 2)
dx
ex
x
dy
1
26.
= ex xe ex ln x +
dx
x
24.
dy
1
28.
=
dx
3
29.
r
3
sec2 x tan4 x
ex arcsin x
dy
x5 e−4x ln3 2x
=
dx
sin2 3x cos4 5x
x
dy
1
= xx xx ln x ln x + 1 +
dx
x ln x
dy
2
27.
= x2 e3x 45x 67x
+ 3 + 5 ln 4 + 7 ln 6
dx
x
25.
4 sec2 x
1
2 tan x +
−1− √
2
tan x
1 − x arcsin x
5
3
−4+
− 6 cot 3x + 20 tan 5x
x
x ln 2x