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Natural Graph of Algebraic Properties of ln(x) Limits Extending the of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives summaries exp(x) = inverse of ln(x)

Last day, we saw that the function f (x) = ln x is one-to-one, with domain (0, ∞) and range (−∞, ∞). We can conclude that f (x) has an which we call the natural and denote (temorarily) by f −1(x) = exp(x), The definition of inverse functions gives us the following:

y = f −1(x) if and only if x = f (y)

y = exp(x) if and only if x = ln(y) The cancellation laws give us:

f −1(f (x)) = x and f (f −1(x)) = x

exp(ln x) = x and ln(exp(x)) = x .

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Graph of exp(x)

We can draw the graph of y = exp(x) by reflecting the graph of y = ln(x) in the line y = x.

have that the graph y = exp(x) is one-to-one and continuous with 20 domain (−∞, ∞) and range (0, ∞). Note that exp(x) > 0 for all values of x. We see that x 15 y = expHxL = e exp(0) = 1 since ln 1 = 0 exp(1) = e since ln e = 1, exp(2) = e2 since ln(e2) = 2, 10 exp(−7) = e−7 since ln(e−7) = −7. 2 H2, e In fact for any r, we have 5 exp(r) = er since ln(er ) = r ln e = H1, eL = H L y ln x r, 2 H-7, e-7L H0, 1L He , 2L He, 1L by the laws of . H L -5 1, 0 5 10

-5

He-7Annette, -7L Pilkington Natural Logarithm and Natural Exponential

-10 We Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Definition of ex .

Definition When x is rational or irrational, we define ex to be exp(x).

ex = exp(x)

Note: This agrees with definitions of ex given elsewhere (as limits), since the definition is the same when x is a rational number and the exponential function is continuous.

Restating the above properties given above in light of this new interpretation of the exponential function, we get: When f (x) = ln(x), f −1(x) = ex and

ex = y if and only if ln y = x

eln x = x and ln ex = x

Annette Pilkington Natural Logarithm and Natural Exponential I Applying the exponential function to both sides of the equation ln(x + 1) = 5, we get eln(x+1) = e5 ln u I Using the fact that e = u, (with u = x + 1 ), we get

x + 1 = e5, or x = e5 − 1 .

I Applying the natural logarithm function to both sides of the equation ex−4 = 10, we get ln(ex−4) = ln(10) u I Using the fact that ln(e ) = u, (with u = x − 4) , we get x − 4 = ln(10), or x = ln(10) + 4.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Solving Equations

We can use the below to solve equations involving logarithms and exponentials. eln x = x and ln ex = x Example Solve for x if ln(x + 1) = 5

Example Solve for x if ex−4 = 10

Annette Pilkington Natural Logarithm and Natural Exponential ln u I Using the fact that e = u, (with u = x + 1 ), we get

x + 1 = e5, or x = e5 − 1 .

I Applying the natural logarithm function to both sides of the equation ex−4 = 10, we get ln(ex−4) = ln(10) u I Using the fact that ln(e ) = u, (with u = x − 4) , we get x − 4 = ln(10), or x = ln(10) + 4.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Solving Equations

We can use the formula below to solve equations involving logarithms and exponentials. eln x = x and ln ex = x Example Solve for x if ln(x + 1) = 5

I Applying the exponential function to both sides of the equation ln(x + 1) = 5, we get eln(x+1) = e5

Example Solve for x if ex−4 = 10

Annette Pilkington Natural Logarithm and Natural Exponential I Applying the natural logarithm function to both sides of the equation ex−4 = 10, we get ln(ex−4) = ln(10) u I Using the fact that ln(e ) = u, (with u = x − 4) , we get x − 4 = ln(10), or x = ln(10) + 4.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Solving Equations

We can use the formula below to solve equations involving logarithms and exponentials. eln x = x and ln ex = x Example Solve for x if ln(x + 1) = 5

I Applying the exponential function to both sides of the equation ln(x + 1) = 5, we get eln(x+1) = e5 ln u I Using the fact that e = u, (with u = x + 1 ), we get

x + 1 = e5, or x = e5 − 1 .

Example Solve for x if ex−4 = 10

Annette Pilkington Natural Logarithm and Natural Exponential u I Using the fact that ln(e ) = u, (with u = x − 4) , we get x − 4 = ln(10), or x = ln(10) + 4.

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Solving Equations

We can use the formula below to solve equations involving logarithms and exponentials. eln x = x and ln ex = x Example Solve for x if ln(x + 1) = 5

I Applying the exponential function to both sides of the equation ln(x + 1) = 5, we get eln(x+1) = e5 ln u I Using the fact that e = u, (with u = x + 1 ), we get

x + 1 = e5, or x = e5 − 1 .

Example Solve for x if ex−4 = 10

I Applying the natural logarithm function to both sides of the equation ex−4 = 10, we get ln(ex−4) = ln(10)

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Solving Equations

We can use the formula below to solve equations involving logarithms and exponentials. eln x = x and ln ex = x Example Solve for x if ln(x + 1) = 5

I Applying the exponential function to both sides of the equation ln(x + 1) = 5, we get eln(x+1) = e5 ln u I Using the fact that e = u, (with u = x + 1 ), we get

x + 1 = e5, or x = e5 − 1 .

Example Solve for x if ex−4 = 10

I Applying the natural logarithm function to both sides of the equation ex−4 = 10, we get ln(ex−4) = ln(10) u I Using the fact that ln(e ) = u, (with u = x − 4) , we get x − 4 = ln(10), or x = ln(10) + 4.

Annette Pilkington Natural Logarithm and Natural Exponential I As it stands, this has an since both numerator and denominator approach infinity as x → ∞

I We modify a trick from 1 and divide (both Numertor and denominator) by the highest of ex in the denominator.

ex ex /ex lim = lim x→∞ 10ex − 1 x→∞ (10ex − 1)/ex

I 1 1 = lim = x→∞ 10 − (1/ex ) 10

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Limits

From the graph we see that

lim ex = 0, lim ex = ∞. x→−∞ x→∞

ex Example Find the limit limx→∞ 10ex −1 .

Annette Pilkington Natural Logarithm and Natural Exponential I We modify a trick from Calculus 1 and divide (both Numertor and denominator) by the highest power of ex in the denominator.

ex ex /ex lim = lim x→∞ 10ex − 1 x→∞ (10ex − 1)/ex

I 1 1 = lim = x→∞ 10 − (1/ex ) 10

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Limits

From the graph we see that

lim ex = 0, lim ex = ∞. x→−∞ x→∞

ex Example Find the limit limx→∞ 10ex −1 . I As it stands, this limit has an indeterminate form since both numerator and denominator approach infinity as x → ∞

Annette Pilkington Natural Logarithm and Natural Exponential I 1 1 = lim = x→∞ 10 − (1/ex ) 10

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Limits

From the graph we see that

lim ex = 0, lim ex = ∞. x→−∞ x→∞

ex Example Find the limit limx→∞ 10ex −1 . I As it stands, this limit has an indeterminate form since both numerator and denominator approach infinity as x → ∞

I We modify a trick from Calculus 1 and divide (both Numertor and denominator) by the highest power of ex in the denominator.

ex ex /ex lim = lim x→∞ 10ex − 1 x→∞ (10ex − 1)/ex

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Limits

From the graph we see that

lim ex = 0, lim ex = ∞. x→−∞ x→∞

ex Example Find the limit limx→∞ 10ex −1 . I As it stands, this limit has an indeterminate form since both numerator and denominator approach infinity as x → ∞

I We modify a trick from Calculus 1 and divide (both Numertor and denominator) by the highest power of ex in the denominator.

ex ex /ex lim = lim x→∞ 10ex − 1 x→∞ (10ex − 1)/ex

I 1 1 = lim = x→∞ 10 − (1/ex ) 10

Annette Pilkington Natural Logarithm and Natural Exponential I 2 2 ex e2x+1 ex +2x+1 = (ex )2 e2x

I 2 2 = ex +2x+1−2x = ex +1

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Rules of exponentials

The following rules of exponents follow from the rules of logarithms:

ex ex+y = ex ey , ex−y = , (ex )y = exy . ey

Proof see notes for details

2 ex e2x+1 Example Simplify (ex )2 .

Annette Pilkington Natural Logarithm and Natural Exponential I 2 2 = ex +2x+1−2x = ex +1

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Rules of exponentials

The following rules of exponents follow from the rules of logarithms:

ex ex+y = ex ey , ex−y = , (ex )y = exy . ey

Proof see notes for details

2 ex e2x+1 Example Simplify (ex )2 .

I 2 2 ex e2x+1 ex +2x+1 = (ex )2 e2x

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Rules of exponentials

The following rules of exponents follow from the rules of logarithms:

ex ex+y = ex ey , ex−y = , (ex )y = exy . ey

Proof see notes for details

2 ex e2x+1 Example Simplify (ex )2 .

I 2 2 ex e2x+1 ex +2x+1 = (ex )2 e2x

I 2 2 = ex +2x+1−2x = ex +1

Annette Pilkington Natural Logarithm and Natural Exponential I Using the , we get

d 2 2 d esin x = esin x · sin2 x dx dx

I 2 2 = esin x 2(sin x)(cos x) = 2(sin x)(cos x)esin x

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Derivatives

d d ex = ex eg(x) = g 0(x)eg(x) dx dx

Proof We use logarithmic differentiation. If y = ex , we have ln y = x and 1 dy dy x differentiating, we get y dx = 1 or dx = y = e . The on the right follows from the chain rule.

d sin2 x Example Find dx e

Annette Pilkington Natural Logarithm and Natural Exponential I 2 2 = esin x 2(sin x)(cos x) = 2(sin x)(cos x)esin x

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Derivatives

d d ex = ex eg(x) = g 0(x)eg(x) dx dx

Proof We use logarithmic differentiation. If y = ex , we have ln y = x and 1 dy dy x differentiating, we get y dx = 1 or dx = y = e . The derivative on the right follows from the chain rule.

d sin2 x Example Find dx e I Using the chain rule, we get

d 2 2 d esin x = esin x · sin2 x dx dx

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Derivatives

d d ex = ex eg(x) = g 0(x)eg(x) dx dx

Proof We use logarithmic differentiation. If y = ex , we have ln y = x and 1 dy dy x differentiating, we get y dx = 1 or dx = y = e . The derivative on the right follows from the chain rule.

d sin2 x Example Find dx e I Using the chain rule, we get

d 2 2 d esin x = esin x · sin2 x dx dx

I 2 2 = esin x 2(sin x)(cos x) = 2(sin x)(cos x)esin x

Annette Pilkington Natural Logarithm and Natural Exponential I Using the chain rule, we get

d 2 2 d 2 sin2(ex ) = 2 sin(ex ) · sin(ex ) dx dx

I 2 2 d 2 = 2 sin(ex ) cos(ex ) · ex dx I 2 2 2 d 2 2 2 = 2 sin(ex ) cos(ex )ex · x 2 = 4xex sin(ex ) cos(ex ) dx

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Derivatives

d d ex = ex eg(x) = g 0(x)eg(x) dx dx

d 2 x2 Example Find dx sin (e )

Annette Pilkington Natural Logarithm and Natural Exponential I 2 2 d 2 = 2 sin(ex ) cos(ex ) · ex dx I 2 2 2 d 2 2 2 = 2 sin(ex ) cos(ex )ex · x 2 = 4xex sin(ex ) cos(ex ) dx

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Derivatives

d d ex = ex eg(x) = g 0(x)eg(x) dx dx

d 2 x2 Example Find dx sin (e ) I Using the chain rule, we get

d 2 2 d 2 sin2(ex ) = 2 sin(ex ) · sin(ex ) dx dx

Annette Pilkington Natural Logarithm and Natural Exponential I 2 2 2 d 2 2 2 = 2 sin(ex ) cos(ex )ex · x 2 = 4xex sin(ex ) cos(ex ) dx

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Derivatives

d d ex = ex eg(x) = g 0(x)eg(x) dx dx

d 2 x2 Example Find dx sin (e ) I Using the chain rule, we get

d 2 2 d 2 sin2(ex ) = 2 sin(ex ) · sin(ex ) dx dx

I 2 2 d 2 = 2 sin(ex ) cos(ex ) · ex dx

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Derivatives

d d ex = ex eg(x) = g 0(x)eg(x) dx dx

d 2 x2 Example Find dx sin (e ) I Using the chain rule, we get

d 2 2 d 2 sin2(ex ) = 2 sin(ex ) · sin(ex ) dx dx

I 2 2 d 2 = 2 sin(ex ) cos(ex ) · ex dx I 2 2 2 d 2 2 2 = 2 sin(ex ) cos(ex )ex · x 2 = 4xex sin(ex ) cos(ex ) dx

Annette Pilkington Natural Logarithm and Natural Exponential 2 I Using substitution, we let u = x + 1. du du = 2x dx, = x dx 2

I Z 2 Z du 1 Z 1 xex +1dx = eu = eudu = eu + 2 2 2

I Switching back to x, we get

1 2 = ex +1 + C 2

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Integrals

Z Z ex dx = ex + C g 0(x)eg(x)dx = eg(x) + C

2 Example Find R xex +1dx.

Annette Pilkington Natural Logarithm and Natural Exponential I Z 2 Z du 1 Z 1 xex +1dx = eu = eudu = eu + C 2 2 2

I Switching back to x, we get

1 2 = ex +1 + C 2

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Integrals

Z Z ex dx = ex + C g 0(x)eg(x)dx = eg(x) + C

2 Example Find R xex +1dx. 2 I Using substitution, we let u = x + 1. du du = 2x dx, = x dx 2

Annette Pilkington Natural Logarithm and Natural Exponential I Switching back to x, we get

1 2 = ex +1 + C 2

Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Integrals

Z Z ex dx = ex + C g 0(x)eg(x)dx = eg(x) + C

2 Example Find R xex +1dx. 2 I Using substitution, we let u = x + 1. du du = 2x dx, = x dx 2

I Z 2 Z du 1 Z 1 xex +1dx = eu = eudu = eu + C 2 2 2

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Integrals

Z Z ex dx = ex + C g 0(x)eg(x)dx = eg(x) + C

2 Example Find R xex +1dx. 2 I Using substitution, we let u = x + 1. du du = 2x dx, = x dx 2

I Z 2 Z du 1 Z 1 xex +1dx = eu = eudu = eu + C 2 2 2

I Switching back to x, we get

1 2 = ex +1 + C 2

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Summary of

ln(x) ex

a x ln(x) ln(ab) = ln a+ln b, ln( ) = ln a−ln b ln e = x and e = x b x ln a = x ln a ex ex+y = ex ey , ex−y = , (ex )y = exy . ey

lim ln x = ∞, lim ln x = −∞ x x x→∞ x→0 lim e = ∞, and lim e = 0 x→∞ x→−∞ 0 d 1 d g (x) d d ln |x| = , ln |g(x)| = ex = ex , eg(x) = g 0(x)eg(x) dx x dx g(x) dx dx Z 1 Z dx = ln |x| + C x x x e dx = e + C Z g 0(x) Z dx = ln |g(x)| + C. g 0(x)eg(x)dx = eg(x) + C g(x)

Annette Pilkington Natural Logarithm and Natural Exponential Natural Logarithm Function Graph of Natural Logarithm Algebraic Properties of ln(x) Limits Extending the antiderivative of 1/x Differentiation and integration Logarithmic differentiation Exponentials Graph ex Solving Equations Limits Laws of Exponentials Derivatives Derivatives Integrals summaries Summary of methods

Logarithmic Differentiation Solving equations (Finding formulas for inverse functions) Finding of inverse functions (using formula from lecture 1). Calculating Limits Calculating Derivatives Calculating Integrals (including definite integrals)

Annette Pilkington Natural Logarithm and Natural Exponential