Curve Tracing of Exponential and
Logarithmic Functions
CALENG1
Dr. Susan A. Roces
2T, AY 2024-2025
Illustrative Examples:
x
e
1. y =
x
Steps:
1. Domain
D : (−∞, 0) U (0,∞)
2. Symmetry
with respect to x-axis
let : y → − y
x
e
−y=
x
y
−y
not symmetrical wrt x − axis
wrt y-axis:
let : x → − x :
−x
−x
x
e
y=
−x
not symmetrical wrt y − axis
wrt (0,0):
let : x → − x
y → −y
y
−x
x
−x
−y
e
−y=
−x
not symmetrical wrt (0,0)
3. Intercepts
x-intercept: let : y = 0
x
e
0=
x
(a,0)
x
e =0
x
ln e = ln 0
x = −∞
no x-intercept
y-intercept: let : x = 0
0
e
y=
0
(0, a )
1
y=
0
y = ±∞
no y-intercept
4. Asymptote
4 a). Horizontal
Asymptote
lim y = a
x→ ±∞
y = a, H . A.
y=a
Note:
Solve for point of intersection of the H. A.
with the curve.
a).
x
∞
e
e
∞
lim
=
=
x→ + ∞ x
∞ ∞
x
e
∞
= lim
=e = ∞
x→ + ∞ 1
No horizontal asymptote
as x → + ∞, y → + ∞
b).
x
e e−∞
1
lim
=
=
∞
x→ − ∞ x
−∞
(e )(−∞)
1
=
= 0
−∞
y = 0, H . A.
as x → − ∞, y → 0
Intersection of the curve with H. A.
x
e
y=
x
x
e
0=
x
x
e =0
x
ln e = ln 0
x = −∞
and y = 0
Therefore:
No
intersection
4 b). Vertical Asymptote
Definition:
lim y = ± ∞
x→ a
x = a, V . A.
To solve for the VA:
Equate the
denominator to
zero!
x=a
x
e
lim
=±∞
x→ a x
Equate the denominator to zero!
x = 0, V . A.
Note:
Take the one-sided
limit when x=0.
x
x
e
a) lim+
x→ 0
x
e
b) lim−
x→ 0
x
0
+
−
0
e
=
0.0001
1
= = +∞
0
x → 0 , y → +∞
e
=
− 0.0001
1
= − = −∞
0
x → 0 , y → −∞
5. Region
Ø based the region by x-intercept and
vertical asymptote
VA
(a,0) III
x − axis
II
x=b x=a
I : x 〈 b : x = __ : y = +
II : b 〈 x 〈 a : x = __ : y = −
III : x 〉 a : x = __ : y = ±
I
II
I
x=0
I (−∞, 0) :
(0,∞) :
II
x = −1 :
x= 2 :
−1
y =e / −1 = −
2
y =e / 2
=+
6. Sketch the curve
x → +∞, y → + ∞
y=0
+
x → 0 , y → +∞
x → − ∞, y → 0
−
x → 0 , y → −∞
x=0
x
e −2
2. y = x
e
Steps:
1. Domain: (−∞, + ∞)
2. Symmetry
wrt x-axis:
let : y → − y
x
e −2
−y= x
e
Not symmetrical wrt
x-axis
wrt y-axis:
−x
let : x → − x
e − 2 Not symmetrical
y = −x
wrt y-axis
e
wrt (0,0): let : x → − x; y → − y
−x
e −2
− y = −x
Not symmetrical
e
wrt (0,0)
3. Intercepts
x-intercept: let : y = 0
x
e −2
0= x
e
x
e −2 =0
x
e =2
x ln e = ln 2
x = ln 2
x-intercept:
P (ln 2,0)
y-intercept: let : x = 0
0
e −2
y= 0
e
1− 2
y=
1
−1
y=
= −1
1
y-intercept:
P (0,−1)
4. Asymptote
4a). Horizontal Asymptote
∞
x
e −2
e −2
∞
= ∞
a) lim x
=
x→∞
e
e
∞
x
e
= lim x = lim 1 = 1
x→∞ e
x→∞
Therefore:
y = 1,
H . A.
as x → + ∞, y →1
Intersection of the curve with H. A.
x
e −2
y
=
1
y=
and
x
e
x
e −2
Therefore:
1=
x
e
x
x
No
e = e −2
intersection
x
x
e −e = −2
x
e − 2 e−∞ − 2
b). lim
=
x
x →−∞
−∞
e
e
1
−2
0−2 −2
∞
=
=
=
−
∞
=
1
0
0
∞
as x → − ∞, y →− ∞
4b). Vertical Asymptote
x
e −2
y= x
e
Equate the den. to zero:
x
e =0
x
ln e = ln 0
x =− ∞
Therefore:
No VA
5. Region
II
I
x = ln 2
0
e −2
I : (−∞, ln 2) : x = 0 : y = 0 = −
e
2
e −2
II : (ln 2,∞) : x = 2 : y = 2 = +
e
6. Sketch the Curve
x → + ∞, y →1
y =1
(ln 2,0)
(0,−1)
x → − ∞, y →− ∞
x = ln 2
Logarithmic Curve
y = ln x
ln 0.2 = −1.6
ln 0.5 = −0.69
ln1.5 = + 0.4
ln 5 = + 1.6
(e,1)
−N
−N
N 〉1
N 〈 1 (1,0)
+N
+N
N 〈1
N 〉1
x = 0.4 x = 1
x=e
3. y = ( x − 2) ln ( x − 5)
Steps:
1. Domain
(5,∞)
2. Symmetry
let : y → − y
wrt x-axis:
− y = ( x − 2) ln ( x − 5)
No
let : x → − x
y = (− x − 2) ln (− x − 5)
No
wrt y-axis:
wrt (0,0): let : x → − x, y → − y
− y = (− x − 2) ln (− x − 5)
y = ( x + 2) ln (− x − 5) No
3. Intercepts
x-int: let: y=0
0 = ( x − 2) ln ( x − 5)
P (2,0)
x−2=0 x = 2
ln ( x − 5) = 0
e
ln( x −5 )
0
=e
x −5 =1
x = 6 P (6,0)
y-int: let: x=0
y = (0 − 2) ln (0 − 5)
no y − int ercept
4. Asymptote
4a). Horizontal Asymptote
a). lim (x − 2)ln (x − 5)
x→ + ∞
= ∞(∞) = ∞
as x → + ∞, y →+∞
b). lim (x − 2)ln (x − 5)
x→ − ∞
= −∞(DNE) = DNE
4b). Vertical Asymptote
ln
(
a
−
a
)
Note: ln ( x − a )
( x − a) = 0
x = a, VA
= ln 0
= −∞
y = ( x − 2) ln ( x − 5)
VA : x − 5 = 0
x=5
a). lim ( x − 2) ln ( x − 5)
x→ 5
+
= (5 − 2) ln (5.001 − 5) = (3) ln (0.001)
= (3)(−∞) = −∞
+
as x → 5 ,
y → −∞
b). lim ( x − 2) ln ( x − 5)
x→ 5 −
= (5 − 2) ln (4.999 − 5)
= (3) ln (−0.001)
= (3) ( DNE )
= DNE
5. Region
I
II
III
IV
x=2
x
I (−∞, 2) : 0
II (2, 5) : 4
x=6
x=5
y = ( x − 2) ln( x − 5)
(0 − 2) ln (0 − 5) = DNE
(4 − 2) ln (4 − 5) = DNE
III (5, 6) : 5.5 (5.5 − 2) ln (5.5 − 5) = +(−) = −
(10 − 2) ln (10 − 5) = +(+) = +
IV (6,∞) : 10
6. Sketch the curve
x → + ∞, y →+∞
(6,0)
+
x=2
x → 5 , y →−∞
x=5 x=6
ln x − 1
4. y =
x−2
Steps:
1. Domain (0, 2) U (2,∞)
2. Symmetry
wrt x-axis: let : y → − y
ln x − 1
−y=
x−2
No
wrt y-axis: let : x → − x
ln(− x) − 1
y=
No
− x−2
wrt (0,0): let : x → − x, y → − y
ln(− x) − 1
No
−y=
− x−2
3. Intercepts
x-int: let: y = 0
y-int: let: x = 0
ln x − 1
ln
0
−
1
0=
y=
x−2
0−2
ln x − 1 = 0
y = −∞
ln x = 1
No
y-intercept
ln x
1
e =e
x = e P(e,0)
4a). Horizontal Asymptote
ln x − 1 ∞
a). lim
=
x→ +∞ x − 2
∞
1
−0
1
x
= lim
=
=0
x→ +∞ 1− 0
∞
y = 0, HA as x → ∞, y → 0
Intersection of the curve with HA:
ln x − 1
y=
and y = 0
x−2
ln x − 1
=0
x−2
ln x − 1 = 0
x= e
P(e,0)
ln x − 1
b). lim
= DNE
x→ −∞ x − 2
4b). Vertical Asymptote
Equate the denominator to zero:
x−2=0
x = 2, VA
ln x: x = 0, VA
Note:
Take one-sided limits
when x = 2 and x = 0
when x = 2:
ln x − 1
a). lim
x→ 2 + x − 2
ln 2 − 1
=
2.0001− 2
+0.N − 1
=
0.0001
−N
=
= −∞
0 +
as x → 2 , y → −∞
ln x − 1
b). lim
x→ 2 − x − 2
ln 2 − 1
=
1.999 − 2
+ 0.N − 1
=
− 0.0001
−N
=
=∞
−0 −
as x → 2 , y → +∞
when x = 0:
ln x − 1
a). lim
x→ 0 + x − 2
+
ln 0 − 1
=
0−2
−∞ − 1
= +∞
=
−2
+
as x → 0 , y → +∞
ln x − 1
b). lim
x→ 0 − x − 2
−
ln 0 − 1
=
0−2
DNE − 1
= DNE
=
−2
as x → 0 , y → DNE
−
5. Region
I
II
III
IV
x =0 x=2 x=e
y = (ln x − 1) /( x − 2)
x
I (−∞, 0) − 1
II (0, 2) : 1
[ln( −1) − 1] /(−1 − 2) = DNE
(ln 1 − 1) /(1 − 2) = − / − = +
III (2, e) : 2.5 (ln 2.5 − 1) /(2.5 − 2)= − / + = −
(ln 4 − 1) /(4 − 2) = + / + = +
IV (e,∞) : 4
6. Trace
(e,0)
y=0
x=0
x=2 x=e
ln x
5. y =
x
( x − 4) e
Steps:
1. Domain
(0, 4) U (4,∞)
2. Symmetry
wrt x-axis:
let : y → − y
ln x
−y=
x
( x − 4) e
Not symmetrical wrt
x-axis
wrt y-axis: let : x → − x
ln(− x)
y=
No
−x
( − x − 4) e
wrt (0,0): let : x → − x, y → − y
ln(− x)
No
−y=
−x
(− x − 4) e
3. Intercepts
x-int: let: y = 0
ln x
0=
x
( x − 4) e
ln x = 0
ln x
0
e =e
x=1
P (1,0)
y-int: let: x = 0
ln 0
y=
0
(0 − 4) e
y = DNE
No y-intercept
4a). Horizontal Asymptote
ln x
∞
a) lim
=
x→+∞ (x − 4)e x
∞
1
1
0
x
∞
= lim
=
=
x
x
∞+∞ ∞
x→ +∞ (x − 4)e + e
=0
y = 0, HA as x → ∞, y → 0
Intersection of the curve with HA:
ln x
y=
y=0
x and
( x − 4) e
ln x
0=
x
( x − 4) e
ln x = 0
x = 1 P (1,0)
ln x
b) lim
x→− ∞ (x − 4)e x
= DNE
4b). Vertical Asymptote
Equate the denominator to zero:
x
x−4=0
e =0
x
ln e = ln 0
x = 4, VA
x = DNE
ln x:
x = 0, VA
Note: Take one-sided limits
when x = 4 and x = 0
when x = 4:
ln x
a ) lim+
x
x → 4 ( x − 4) e
ln 4
=
4
(4.001 − 4)e
+N
=
0.0001
= +∞
+
as x → 4 , y → +∞
ln x
b) lim−
x
x → 4 ( x − 4) e
ln 4
=
4
(3.999 − 4)e
+N
=
− 0.0001
= −∞
−
as x → 4 , y → −∞
when x = 0:
ln x
a ) lim+
x
x → 0 ( x − 4) e
ln 0
=
0
(0 − 4)e
−∞
=
−4
= +∞
+
as x → 0 , y → +∞
ln x
b) lim−
x
x → 0 ( x − 4) e
−
ln 0
=
0
(0 − 4)e
DNE
= DNE
=
−4
5. Region
I
II
x=0
III
x =1
IV
x=4
x
x
y = ln x /( x − 4) e
−1
(−∞,
0)
:
− 1 ln(−1) /(−1 − 4) e = DNE
I
0.5
II (0,1) 0.5 ln(0.5) /(0.5 − 4) e = − / − = +
III (1, 4) :
3
IV (4,∞) : 6
3
ln 3 /(3 − 4) e = + / − = −
6
ln 6 /(6 − 4) e = + / + = +
6. Sketch the curve
(1,0)
y=0
x=0
x =1
x=4
2x
(x − 3)e
6. y =
(ln x) − 2
Steps:
(0, e 2 ) U (e 2 ,∞)
1. Domain
2. Symmetry
wrt x-axis:
2x
(x − 3)e
−y =
(ln x) − 2
let : y → − y
Not symmetrical wrt
x-axis
wrt y-axis: let : x → − x
−2 x
(−x − 3)e
y=
No
ln (−x) − 2
wrt (0,0): let : x → − x, y → − y
−2 x
(−x − 3)e
−y =
ln (−x) − 2
No
3. Intercepts
x-int: let: y = 0
2x
(x − 3)e
0=
(ln x) − 2
2x
(x − 3)e = 0
2x
x − 3 = 0, e = 0
x = 3 ln e 2 x = ln 0
P(3,0)
2x = −∞
y-int: let: x = 0
2(0)
(0 − 3)e
y=
(ln 0) − 2
−3
3
y=
= =0
−∞ − 2 ∞
P(0, 0)
Missing Point
4a). Horizontal Asymptote
2x
∞
(x − 3)e
=
a) lim
∞
x→+∞ (ln x) − 2
∞
2(x − 3)e + e
= lim ___________
=
=
∞
x→ +∞
1
0
2x
2x
x
no HA, as x → ∞, y → ∞
2x
(x − 3)e
b) lim
x→− ∞ (ln x) − 2
2(− ∞)
(− ∞ − 3)e
=
(ln − ∞) − 2
= DNE
4b). Vertical Asymptote
Equate the denominator to zero:
(ln x) − 2 = 0
ln x = 2
2
x = e , VA
ln (x): x = 0,
not a VA
Missin g Po int Discontinuity
Note:
Take one-sided limits when x = e2, 0
when x = e2:
2x
(x − 3)e
a) lim
x→e2 + (ln x) − 2
2x
(x − 3)e
b) lim
x→e2 − (ln x) − 2
2
2e2
(e − 3)e
=
2 +
(ln e ) − 2
(e − 3)e
=
2 −
(ln e ) − 2
+N
=
+0
+N
=
−0
=∞
= −∞
2
2e
2 +
2
as x → e , y → ∞
as x → e 2 −, y → −∞
when x = 0:
2x
(x − 3)e
a) lim
x→0+ (ln x) − 2
(−N )(+N )
=
−∞ − 2
−N
=
=0
−∞
+
as x → 0 , y → 0
P(0, 0), Missin g Po int
2x
(x − 3)e
b) lim
x→0− (ln x) − 2
(−N )(+N )
=
DNE − 2
= DNE
−
as x → 0 , y → DNE
x = 0, Missin g Po int Disc.
5. Region
I
II
III
IV
2
x=0 x=3
x=e
2x
x y =(x − 3)e / (ln x) − 2
I x 〈 0:
−1
y = (−N )(+N ) / DNE − 2 = DNE
II 0〈x 〈3 :
1
y = (−N )(+N ) / 0 − 2 = − / − = +
=+/−=−
2
y
=
(+N
)(+N
)
/+
N
−
2
5
3〈x
〈e
:
III
IV
x 〉 e2 :
10 y = (+N )(+N ) /+ N
= +/+ = +
6. Sketch the function
x → e 2 +, y → ∞
x → ∞, y → ∞
(3,0)
P(0, 0), Missin g Po int
P(3, 0), x − int.
x=0
x=3 x=e
2
x → e 2 −, y → −∞
ln (x − 2)
7. y =
x +1
e
Steps:
1. Domain
2. Symmetry
wrt x-axis:
(2,∞)
let : y → − y
ln ( x − 2) Not symmetrical wrt
−y=
x +1
x-axis
e
wrt y-axis: let : x → − x
ln (− x − 2)
No
y=
− x +1
e
wrt (0,0): let : x → − x, y → − y
ln (− x − 2)
No
−y=
− x +1
e
3. Intercepts
x-int: let: y = 0
ln ( x − 2)
0=
x +1
e
ln ( x − 2) = 0
e
ln ( x − 2 )
=e
x− 2 =1
x= 3
P(3,0)
0
y-int: let: x = 0
ln (0 − 2)
y=
0 +1
e
y = DNE
No y-intercept
4a). Horizontal Asymptote
ln (x − 2) ∞
=
a) lim
x +1
∞
e
x→+∞
1
x
−
2
= lim
x +1
1
0
=∞ =
=0
∞
∞
e
y = 0, HA as x → ∞, y → 0
x→ +∞
Intersection of the curve with HA:
ln ( x − 2)
and
y
=
0
y=
x +1
e
ln ( x − 2)
0=
x +1
e
ln ( x − 2) = 0
x = 3 P(3,0)
ln (x − 2)
=
DNE
b) lim
x +1
e
x→− ∞
4b). Vertical Asymptote
Equate the denominator to zero:
x +1
e =0
( x +1)
ln e
= ln 0
x = DNE
ln (x-2):
x−2=0
x
=
2
,
VA
Note:
Take one-sided limits when x = 2
when x = 2:
ln ( x − 2)
a) lim
x +1
+
e
x→ 2
ln (2.0001 − 2)
=
2 +1
e
−∞
=
+N
= −∞
+
as x → 2 , y → −∞
ln ( x − 2)
b) lim
x +1
−
e
x→ 2
ln (1.999 − 2)
=
2 +1
e
DNE
=
+N
= DNE
as x → 2 , y → DNE
−
5. Region
I
II
x=2
x
I x〈2: 0
III
x=3
y = ln ( x − 2) / e
ln(0 − 2) / e
0+1
x +1
= DNE
II 2〈x 〈3 : 2.5 ln(2.5 − 2) / e = − / + = −
5+1
III x 〉 3 : 5 ln(5 − 2) / e = + / + = +
2.5+1
6. Sketch the function
(3,0)
y=0
x=2
x=3
Practice Numbers:
1. y= x ln x
ln x
2. y =
x
2
3. y = x e
− x2
1− ln x
4. y =
2
x
Assignment 23: Curve Tracing of Exponential and
Logarithmic Functions
x
e −2
1. y = x
e
x
2. y = e ln x
3. y = x e
x
2
4. y = (x − 2)ln x
1
x
e
5. y = 2
x
x
xe
6. y =
(ln x) −1