Overview of ex and ln x Functions
• Graphs
• Function Properties
Natural Exponential Function y = ex Natural Logarithm Function y = ln x
• Domain = R
• Domain = (0, ∞)
• Range = (0, ∞)
• Range = R
• lim ex = ∞
• lim ln x = ∞
x→∞
•
x→∞
lim ex = 0
• lim ln x = −∞
x→−∞
x→0+
eh − 1
=1
h→0
h
• e chosen so that lim
• y = ln x is the inverse function for y = ex
• 2 < e < 3 and e ≈ 2.71828 . . .
• y = ln x if and only if ey = x
x
ln e = x for all x in R
¶
µ
1 n
• e = lim 1 +
n→∞
n
•
• e0 = 1 and e1 = e
• ln 1 = 0 and
ln x
e
= x for all x > 0
ln e = 1 • Algebraic Properties
Natural Exponential Function y = ex Natural Logarithm Function y = ln x
• ex ey = ex+y
•
ex−y
ex
= y
e
• ln(xy) = ln x + ln y
µ ¶
x
• ln
= ln x − ln y
y
• (ex )y = exy
• ln(xy ) = y ln x
• e(x
ln(x + y) or
y)
cannot be simplified
ln x
cannot be simplified
ln y
• Derivatives and Integrals
Natural Exponential Function y = ex Natural Logarithm Function y = ln x
•
d x
e = ex
dx
du
d u(x)
e
= eu(x)
= eu(x) u′ (x)
dx
dx
Z
•
ex dx = ex + C
•
•
d
1
ln x =
dx
x
d
1 du
u′ (x)
ln u(x) =
=
dx
u(x) dx
u(x)
Z
1
•
dx = ln |x| + C
x
•