the natural logarithm and exponential functions · 2014. 1. 31. · exponential function...

The NaturalLogarithm andExponentialFunctions

BanderAlmutairi

GeneralExponentialFunction

Derivatives ofGeneralExponentialFunction

Integration ofGeneralExponentialFunctions

GeneralLogarithmFunctions

Derivative ofGeneralLogarithmicFunction

The Natural Logarithm and ExponentialFunctions

Bander Almutairi

King Saud University

26 Sept 2013

BanderAlmutairi

1 General Exponential Function

2 Derivatives of General Exponential Function

3 Integration of General Exponential Functions

4 General Logarithm Functions

5 Derivative of General Logarithmic Function

BanderAlmutairi

Definition

If a is positive, then a function f (x) = ax is called theexponential function with base a and x called the exponent.

The function f (x) is also called general exponential function.Examples: f (x) = 2x , g(x) = 6x .* If the exponent is a rational number r , then

ax = e ln(ar ) = er ln(a), a > 0.

* Relation between general and natural exponential is

ax = ex ln(a), a > 0, x ∈ R.

BanderAlmutairi

Definition

The function f (x) is also called general exponential function.

Examples: f (x) = 2x , g(x) = 6x .* If the exponent is a rational number r , then

ax = e ln(ar ) = er ln(a), a > 0.

ax = ex ln(a), a > 0, x ∈ R.

BanderAlmutairi

Definition

The function f (x) is also called general exponential function.Examples: f (x) = 2x , g(x) = 6x .

* If the exponent is a rational number r , then

ax = e ln(ar ) = er ln(a), a > 0.

ax = ex ln(a), a > 0, x ∈ R.

BanderAlmutairi

Definition

ax = e ln(ar ) = er ln(a), a > 0.

ax = ex ln(a), a > 0, x ∈ R.

BanderAlmutairi

Definition

ax = e ln(ar ) = er ln(a), a > 0.

ax = ex ln(a), a > 0, x ∈ R.

BanderAlmutairi

Definition

ax = e ln(ar ) = er ln(a), a > 0.

ax = ex ln(a), a > 0, x ∈ R.

BanderAlmutairi

Definition

ax = e ln(ar ) = er ln(a), a > 0.

ax = ex ln(a), a > 0, x ∈ R.

BanderAlmutairi

Laws of Exponents:

Let a > 0 and b > 0. If u and v are anyreal numbers

(i) auav = au+v .

(ii) au/av = au−v .

(iii) (au)v = auv .

(iv) (ab)u = aubu.

(v) (a/b)u = au/bu.

BanderAlmutairi

Laws of Exponents: Let a > 0 and b > 0. If u and v are anyreal numbers

(i) auav = au+v .

(iii) (au)v = auv .

(iv) (ab)u = aubu.

(v) (a/b)u = au/bu.

BanderAlmutairi

(i) auav = au+v .

(iii) (au)v = auv .

(iv) (ab)u = aubu.

(v) (a/b)u = au/bu.

BanderAlmutairi

(i) auav = au+v .

(iii) (au)v = auv .

(iv) (ab)u = aubu.

(v) (a/b)u = au/bu.

BanderAlmutairi

(i) auav = au+v .

(iii) (au)v = auv .

(iv) (ab)u = aubu.

(v) (a/b)u = au/bu.

BanderAlmutairi

(i) auav = au+v .

(iii) (au)v = auv .

(iv) (ab)u = aubu.

(v) (a/b)u = au/bu.

BanderAlmutairi

(i) auav = au+v .

(iii) (au)v = auv .

(iv) (ab)u = aubu.

(v) (a/b)u = au/bu.

BanderAlmutairi

Theorem

If a > 0, then

dx(ax) = ax . ln(a).

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

= ex ln(a). ln(a) = ax . ln(a)

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax)

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

= ex ln(a).d

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

= ex ln(a). ln(a)

= ax . ln(a)

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

= 3√2x ln(3)

BanderAlmutairi

Theorem

If a > 0, thend

Proof:

dx(ax) =

(ex ln(a)

)= ex ln(a).

dx(x ln(a))

Examples:

(a) ddx (5x) = 5x ln(5).

(b) ddx

(3√2x)

= 3√2x ln(3) d

dx (√

2x) = 3√2x ln(3)

BanderAlmutairi

* If a > 1,

then ln(a) > 0,therefore,

dx(ax) = ax ln(a) > 0.

Hence f (x) = ax is increasing function on the interval (1,∞).

* If 0 < a < 1, then ln(a) < 0, therefore,

dx(ax) = ax ln(a) < 0.

Hence f (x) = ax is decreasing function on the interval (0, 1)

BanderAlmutairi

* If a > 1, then ln(a) > 0,

therefore,

BanderAlmutairi

* If a > 1, then ln(a) > 0,therefore,

BanderAlmutairi

* If 0 < a < 1,

then ln(a) < 0, therefore,

BanderAlmutairi

* If 0 < a < 1, then ln(a) < 0,

therefore,

BanderAlmutairi

Examples: Find f ′(x) in each of the following:

(a) f (x) = 5√x3 .

f ′(x) = 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x) = 71−2x5

ln(7)d

(1− 2x5

)= 71−2x

5ln(7)(−10x4).

BanderAlmutairi

(a) f (x) = 5√x3 .

f ′(x) = 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x) = 71−2x5

ln(7)d

(1− 2x5

)= 71−2x

5ln(7)(−10x4).

BanderAlmutairi

(a) f (x) = 5√x3 .

f ′(x)

= 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x) = 71−2x5

ln(7)d

(1− 2x5

)= 71−2x

5ln(7)(−10x4).

BanderAlmutairi

(a) f (x) = 5√x3 .

f ′(x) = 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x) = 71−2x5

ln(7)d

(1− 2x5

)= 71−2x

5ln(7)(−10x4).

BanderAlmutairi

(a) f (x) = 5√x3 .

f ′(x) = 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x) = 71−2x5

ln(7)d

(1− 2x5

)= 71−2x

5ln(7)(−10x4).

BanderAlmutairi

(a) f (x) = 5√x3 .

f ′(x) = 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x) = 71−2x5

ln(7)d

(1− 2x5

)= 71−2x

5ln(7)(−10x4).

BanderAlmutairi

(a) f (x) = 5√x3 .

f ′(x) = 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x)

= 71−2x5

ln(7)d

(1− 2x5

)= 71−2x

5ln(7)(−10x4).

BanderAlmutairi

(a) f (x) = 5√x3 .

f ′(x) = 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x) = 71−2x5

ln(7)d

(1− 2x5

= 71−2x5

ln(7)(−10x4).

BanderAlmutairi

(a) f (x) = 5√x3 .

f ′(x) = 5√x3 ln(5)

(√x3)

= 5√x3 ln(5).

2√x3.

(b) f (x) = 71−2x5.

f ′(x) = 71−2x5

ln(7)d

(1− 2x5

)= 71−2x

5ln(7)(−10x4).

BanderAlmutairi

(c) f (x) = (2x + 2−x)9.

f ′(x) = 9(2x + 2−x)8.d

dx(2x + 2−x)

= 9(2x + 2−x)8.

dx(2x) +

dx(2−x)

]= 9(2x + 2−x)8

[2x ln(2)

dx(x) + 2−x ln(2)

dx(−x)

]= 9(2x + 2−x)8

[2x ln(2)− 2−x ln(2)

BanderAlmutairi

(c) f (x) = (2x + 2−x)9.

f ′(x)

= 9(2x + 2−x)8.d

dx(2x + 2−x)

= 9(2x + 2−x)8.

dx(2x) +

dx(2−x)

]= 9(2x + 2−x)8

[2x ln(2)

dx(x) + 2−x ln(2)

dx(−x)

]= 9(2x + 2−x)8

[2x ln(2)− 2−x ln(2)

BanderAlmutairi

(c) f (x) = (2x + 2−x)9.

f ′(x) = 9(2x + 2−x)8.d

dx(2x + 2−x)

= 9(2x + 2−x)8.

dx(2x) +

dx(2−x)

]= 9(2x + 2−x)8

[2x ln(2)

dx(x) + 2−x ln(2)

dx(−x)

]= 9(2x + 2−x)8

[2x ln(2)− 2−x ln(2)

BanderAlmutairi

(c) f (x) = (2x + 2−x)9.

f ′(x) = 9(2x + 2−x)8.d

dx(2x + 2−x)

= 9(2x + 2−x)8.

dx(2x) +

dx(2−x)

= 9(2x + 2−x)8[

2x ln(2)d

dx(x) + 2−x ln(2)

dx(−x)

]= 9(2x + 2−x)8

[2x ln(2)− 2−x ln(2)

BanderAlmutairi

(c) f (x) = (2x + 2−x)9.

f ′(x) = 9(2x + 2−x)8.d

dx(2x + 2−x)

= 9(2x + 2−x)8.

dx(2x) +

dx(2−x)

]= 9(2x + 2−x)8

[2x ln(2)

dx(x) + 2−x ln(2)

dx(−x)

= 9(2x + 2−x)8[2x ln(2)− 2−x ln(2)

BanderAlmutairi

(c) f (x) = (2x + 2−x)9.

f ′(x) = 9(2x + 2−x)8.d

dx(2x + 2−x)

= 9(2x + 2−x)8.

dx(2x) +

dx(2−x)

]= 9(2x + 2−x)8

[2x ln(2)

dx(x) + 2−x ln(2)

dx(−x)

]= 9(2x + 2−x)8

[2x ln(2)− 2−x ln(2)

BanderAlmutairi

Theorem

If a > 0, then

∫audu =

ln(a).au + c .

Proof:We know that d

du (au) = au ln(a). Integrating both sides∫d

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

If a > 0, then ∫audu =

ln(a).au + c .

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

Proof:

We know that ddu (au) = au ln(a). Integrating both sides∫

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du (au)

= au ln(a). Integrating both sides∫d

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du (au) = au ln(a).

Integrating both sides∫d

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du (au) = au ln(a). Integrating both sides

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du(au)du

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

∫audu =

ln(a).au + c .

BanderAlmutairi

Theorem

ln(a).au + c .

du(au)du =

∫au ln(a)du,

so we get

au + c = ln(a)

∫audu.

Hence ∫audu =

ln(a).au + c .

BanderAlmutairi

Examples:

Evaluate the following integrals.[1]∫ 0

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

Put = tan(x), so du = sec2(x)dx = 1cos2(x)

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

Examples: Evaluate the following integrals.

[1]∫ 0

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

Examples: Evaluate the following integrals.[1]∫ 0

−35xdx

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

Put = tan(x),

so du = sec2(x)dx = 1cos2(x)

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

Put = tan(x), so du

= sec2(x)dx = 1cos2(x)

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

Put = tan(x), so du = sec2(x)dx

= 1cos2(x)

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c

ln(5)5tan(x) + c.

BanderAlmutairi

−35xdx =

ln(5)5x]0−3

[1− 5−3

125 ln(5).

[2] ∫5tan(x)

cos2(x)dx .

dx . Hence,

∫5tan(x)

cos2(x)dx =

∫5udu =

ln(5)5u + c =

ln(5)5tan(x) + c.

BanderAlmutairi

∫(3x + 1)2

3xdx =

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

ln(3)+ 2x − 3−x

ln(3)+ c .

BanderAlmutairi

[3] ∫(3x + 1)2

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

ln(3)+ 2x − 3−x

ln(3)+ c .

BanderAlmutairi

[3] ∫(3x + 1)2

3xdx =

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

ln(3)+ 2x − 3−x

ln(3)+ c .

BanderAlmutairi

[3] ∫(3x + 1)2

3xdx =

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

ln(3)+ 2x − 3−x

ln(3)+ c .

BanderAlmutairi

[3] ∫(3x + 1)2

3xdx =

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

ln(3)+ 2x − 3−x

ln(3)+ c .

BanderAlmutairi

[3] ∫(3x + 1)2

3xdx =

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

+ 2x − 3−x

ln(3)+ c .

BanderAlmutairi

[3] ∫(3x + 1)2

3xdx =

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

ln(3)+ 2x

− 3−x

ln(3)+ c .

BanderAlmutairi

[3] ∫(3x + 1)2

3xdx =

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

ln(3)+ 2x − 3−x

BanderAlmutairi

[3] ∫(3x + 1)2

3xdx =

∫ ((3x)2 + 2(3x) + 1

∫ [(3x)2

∫ [3x + 2 + 3−x

ln(3)+ 2x − 3−x

ln(3)+ c .

BanderAlmutairi

Definition

If a 6= 1 and y = ax ,

then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So

y = loga(x)⇐⇒x = ay .

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

If a 6= 1 and y = ax , then the inverse function of y

is denotedby loga(x) and is called logarithm function with base a. So

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x)

and is called logarithm function with base a. So

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a.

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

If a 6= 1 and y = ax , then the inverse function of y is denotedby loga(x) and is called logarithm function with base a. So

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

Examples:

(i) 9 = 32

⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52

⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53

⇐⇒ 3 = log5(125).

BanderAlmutairi

Definition

Examples:

(i) 9 = 32 ⇐⇒ 2 = log3(9).

(ii) 25 = 52 ⇐⇒ 2 = log5(25).

(iii) 125 = 53 ⇐⇒ 3 = log5(125).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

y = logxa ⇒ x = ay ⇒ ln(x) = ln(ay )

⇒ ln(x) = y ln(a)⇒ y =ln(x)

ln(a).

(ii)ln(x) = loge(x).

(iii)log(x) = log10(x)

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

y = logxa

⇒ x = ay ⇒ ln(x) = ln(ay )

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

y = logxa ⇒

x = ay ⇒ ln(x) = ln(ay )

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

y = logxa ⇒ x = ay

⇒ ln(x) = ln(ay )

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

y = logxa ⇒ x = ay ⇒

ln(x) = ln(ay )

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

ln(x) = y ln(a)⇒ y =ln(x)

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

⇒ ln(x) = y ln(a)

⇒ y =ln(x)

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

⇒ ln(x) = y ln(a)⇒

y =ln(x)

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

ln(a).

BanderAlmutairi

loga(x) =ln(x)

ln(a).

Proof:

ln(a).

BanderAlmutairi

Q: Find y ′ in each of the following:

1 y = loga(x).

We know that loga(x) = ln(x)ln(a) = 1

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

We have

y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

We have y

= log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) . Hence,

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

We have y = log(ln(x))

= log10(ln(x)) = ln(ln(x))ln(10) . Hence,

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

We have y = log(ln(x)) = log10(ln(x))

= ln(ln(x))ln(10) . Hence,

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

We have y = log(ln(x)) = log10(ln(x)) = ln(ln(x))ln(10) .

Hence,

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))]

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)]

log(x).

ln(10).1

BanderAlmutairi

1 y = loga(x).

ln(a) ln(x).Thus,

y ′ =1

2 y = log(ln(x)).

y ′ =1

ln(10)

dx[ln(ln(x))] =

ln(10)

3 y = ln(log(x)).

y ′ =1

log(x)

dx[log(x)] =

log(x).

ln(10).1

BanderAlmutairi

Q: Find f ′(x) if f (x) = x4+x2 .

We take ln for both sides, we get

ln[f (x)] = ln[x4+x2

]= (4 + x2) ln(x).

Now we differentiate with respect to x ,

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

Q: Find f ′(x) if f (x) = x4+x2 .We take ln for both sides, we get

]= (4 + x2) ln(x).

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

ln[f (x)]

= ln[x4+x2

]= (4 + x2) ln(x).

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

= (4 + x2) ln(x).

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

]= (4 + x2) ln(x).

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

]= (4 + x2) ln(x).

f (x)f ′(x)

= (4 + x2)d

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

]= (4 + x2) ln(x).

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

]= (4 + x2) ln(x).

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

]= (4 + x2) ln(x).

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x)

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

]= (4 + x2) ln(x).

f (x)f ′(x) = (4 + x2)

dx[ln(x)] + ln(x)

dx[4 + x2]

= (4 + x2)1

x+ ln(x)(2x)

Therefore,

f ′(x) =

[4 + x2

x+ ln(x)(2x)

]x4+x2 .

BanderAlmutairi

Exercises: Find y ′ for the following:

1 y = ππ.

2 y = x4.

3 y = xπ.

4 y = πx .

5 y = x2x .

BanderAlmutairi

1 y = ππ.

2 y = x4.

3 y = xπ.

4 y = πx .

5 y = x2x .

BanderAlmutairi

1 y = ππ.

2 y = x4.

3 y = xπ.

4 y = πx .

5 y = x2x .

BanderAlmutairi

1 y = ππ.

2 y = x4.

3 y = xπ.

4 y = πx .

5 y = x2x .

BanderAlmutairi

1 y = ππ.

2 y = x4.

3 y = xπ.

4 y = πx .

5 y = x2x .

BanderAlmutairi

1 y = ππ.

2 y = x4.

3 y = xπ.

4 y = πx .

5 y = x2x .

the natural logarithm and exponential functions · 2014. 1. 31. · exponential function...

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