Goals
We will
provide a formal, calculus based definition for the natural
logarithm,
use this definition to re-prove logarithmic properties
The Original Definition
logb (a) ≡ x iff b x = a.
Calculate the following.
1
log2 (8) =
2
log3 (9) =
3
log5 (10) =
The Calculus Definition
ln x ≡
Z x
1
1
t
dt.
Properties
ln x ≡
Z x
1
1
t
dt.
Use the definition and derivative and integral knowledge to
calculate the following.
1
ln 1 =
2
d ln x
dx
=
Shape
ln x =
Z x
1
1
t
dt.
What is ln 1?
Remember that an integral adds.
Which is bigger ln 1 or ln 2?
Which is bigger ln 1 or ln a where a > 1?
Which is bigger ln 1 or ln a where a < 1?
Which is bigger ln a or ln b where b > a?
What does this say about the graph of ln x ?
Shape Again
Now we will use integral property to obtain the same result.
What should the relationship between ln(x + a) and ln x be?
Suppose a > 0.
What integral property applies to
Rb
a
f (x ) dx and
Use the definition then simplify ln(x + a) − ln x .
Rc
b
f (x ) dx ?
Product Properties
g(x ) = ln(ax ), f (x ) = ln x .
Calculate f 0 (x ) and g 0 (x ).
Compare f 0 (x ) and g 0 (x ). What does this imply about these
functions and their graphs?
Evaluate your conclusion at x = 1 and substitute this result
into the conclusion.
More Anti-Derivatives
1
Calculate the first derivative of f (x ) = − ln(cos x ).
2
Calculate the first derivative of f (x ) = ln(sec x + tan x ).
3
R
tan x dx
4
R
sec x dx
5
R
csc x dx