The Natural Logarithm Function and The Exponential
Function
One specific logarithm function is singled out and one particular
exponential function is singled out.
Definition
e = limx→0 (1 + x)1/x
Alan H. SteinUniversity of Connecticut
The Natural Logarithm Function and The Exponential
Function
One specific logarithm function is singled out and one particular
exponential function is singled out.
Definition
e = limx→0 (1 + x)1/x
Definition (The Natural Logarithm Function)
ln x = loge x
Alan H. SteinUniversity of Connecticut
The Natural Logarithm Function and The Exponential
Function
One specific logarithm function is singled out and one particular
exponential function is singled out.
Definition
e = limx→0 (1 + x)1/x
Definition (The Natural Logarithm Function)
ln x = loge x
Definition (The Exponential Function)
exp x = e x
Alan H. SteinUniversity of Connecticut
Special Cases
The following special cases of properties of logarithms and
exponential functions are worth remembering separately for the
natural log function and the exponential function.
Alan H. SteinUniversity of Connecticut
Special Cases
The following special cases of properties of logarithms and
exponential functions are worth remembering separately for the
natural log function and the exponential function.
I
y = ln x if and only if x = e y
Alan H. SteinUniversity of Connecticut
Special Cases
The following special cases of properties of logarithms and
exponential functions are worth remembering separately for the
natural log function and the exponential function.
I
y = ln x if and only if x = e y
I
ln(e x ) = x
Alan H. SteinUniversity of Connecticut
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
Alan H. SteinUniversity of Connecticut
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
b x = e x ln b
Alan H. SteinUniversity of Connecticut
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
b x = e x ln b
This important identity is very useful.
Alan H. SteinUniversity of Connecticut
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
b x = e x ln b
This important identity is very useful.
Similarly, suppose y = logb x.
Alan H. SteinUniversity of Connecticut
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
b x = e x ln b
This important identity is very useful.
Similarly, suppose y = logb x. Then, by the definition of a
logarithm, it follows that b y = x.
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
b x = e x ln b
This important identity is very useful.
Similarly, suppose y = logb x. Then, by the definition of a
logarithm, it follows that b y = x. But then ln(b y ) = ln x.
Alan H. SteinUniversity of Connecticut
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
b x = e x ln b
This important identity is very useful.
Similarly, suppose y = logb x. Then, by the definition of a
logarithm, it follows that b y = x. But then ln(b y ) = ln x. Since
ln(b y ) = y ln b,
Alan H. SteinUniversity of Connecticut
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
b x = e x ln b
This important identity is very useful.
Similarly, suppose y = logb x. Then, by the definition of a
logarithm, it follows that b y = x. But then ln(b y ) = ln x. Since
ln x
ln(b y ) = y ln b, it follows that y ln b = ln x and y =
, yielding
ln b
the following equally important identity.
Alan H. SteinUniversity of Connecticut
Other Bases
Suppose y = b x . By the properties of logarithms, we can write
ln y = ln(b x ) = x ln b. It follows that e ln y = e x ln b . But, since
e ln y = y = b x , it follows that
b x = e x ln b
This important identity is very useful.
Similarly, suppose y = logb x. Then, by the definition of a
logarithm, it follows that b y = x. But then ln(b y ) = ln x. Since
ln x
ln(b y ) = y ln b, it follows that y ln b = ln x and y =
, yielding
ln b
the following equally important identity.
logb x =
ln x
ln b
Alan H. SteinUniversity of Connecticut
Derivatives
d
1
(ln x) =
dx
x
Alan H. SteinUniversity of Connecticut
Derivatives
d
1
(ln x) =
dx
x
d
(e x ) = e x
dx
Alan H. SteinUniversity of Connecticut
Derivatives
d
1
(ln x) =
dx
x
d
(e x ) = e x
dx
If there are logs or exponentials with other bases, one may still use
these formulas after rewriting the functions in terms of natural logs
or the exponential function.
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(5x )
dx
Using the formula ax = e x ln a , write 5x as e x ln 5 . We can then
apply the Chain Rule, writing:
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(5x )
dx
Using the formula ax = e x ln a , write 5x as e x ln 5 . We can then
apply the Chain Rule, writing:
y = 5x = e x ln 5
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(5x )
dx
Using the formula ax = e x ln a , write 5x as e x ln 5 . We can then
apply the Chain Rule, writing:
y = 5x = e x ln 5
y = eu
u = x ln 5
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(5x )
dx
Using the formula ax = e x ln a , write 5x as e x ln 5 . We can then
apply the Chain Rule, writing:
y = 5x = e x ln 5
y = eu
u = x ln 5
dy du
dy
=
dx
du dx
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(5x )
dx
Using the formula ax = e x ln a , write 5x as e x ln 5 . We can then
apply the Chain Rule, writing:
y = 5x = e x ln 5
y = eu
u = x ln 5
dy du
dy
=
= e u · ln 5
dx
du dx
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(5x )
dx
Using the formula ax = e x ln a , write 5x as e x ln 5 . We can then
apply the Chain Rule, writing:
y = 5x = e x ln 5
y = eu
u = x ln 5
dy du
dy
=
= e u · ln 5 = 5x ln 5
dx
du dx
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(log7 x)
dx
Using the formula logb x =
ln x
Example: Calculate
d
(log7 x)
dx
ln x
ln x
, we write log7 x =
, so we can
Using the formula logb x =
ln b
ln 7
proceed as follows:
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(log7 x)
dx
ln x
ln x
, we write log7 x =
, so we can
Using the formula logb x =
ln b
ln 7
proceed as follows:
y = log7 x =
1
ln x
=
· ln x
ln 7
ln 7
Alan H. SteinUniversity of Connecticut
Example: Calculate
d
(log7 x)
dx
Using the formula logb x =
ln x