Uniqueness Theorem and
Properties of Log Functions
Lesson 6-3
Logarithm is just a fancy name for exponents.
They were used as a fast way to do calculations
BEFORE calculators were invented.
Exponential and Logarithmic
Properties Correspond:
Product of powers
b b  b
c
d
cd
Quotient of powers
b
b
c
d
b
cd
Power of a power
b 
c d
b
cd
Log of a Product
logb (cd )  logb c  logb d
Log of a Quotient
c
logb    logb c  logb d
d 
Log of a Power
log b (c r )  r log b c
Where b>0, b≠1,c>0,d>0 and r
is any real #
Algebraic Definition of Logarithm
a  logb c if and only if b a  c
a  log10 (100000)
10  100000
a
Since 10  100000
a5
5
Uniqueness Theorem for Derivatives
- if functions start at the same point and
change the same way, they are the same.
- this relies on the Mean Value Theorem
If: 1. f '(x) = g'(x)
for all values of x in
the domain, and
2. f(a) = g(a) for one value, x = a,
in the domain, then f(x) = g(x) for
all values of x in the domain.
This theorem is primarily used in proving that the natural log (ln) has the
properties of logarithms.
Logarithm Properties of Ln:
Product:
Quotient:
Power:
ln(ab)  ln a  ln b
a
ln    ln a  ln b
b
ln(a r )  r ln a
1
Intercept:
1
ln(1)  0 since  dt  0
t
1
Examples:
Evaluate both sides of the equations to show
they are equivalent.
ln(21)  ln 7  ln 3
 28 
ln    ln 28  ln 4
 4 
3.045  1.946  1.099 1.946  3.332  1.386
3.045  3.045
1.946  1.946
Examples:
Evaluate both sides of the equations to show
they are equivalent.
ln(1)  0
ln(6 )  3ln 6
3
5.375  3(1.792)
5.375  5.376
Examples:
For what value of x is ln equal to 1?
ln( x)  1
y  1 @ x  2.7182818
xe
Log in bases other than 10.
Property: Equivalence of Natural Logs and Base
e Logs
ln x  loge x for all x>0
Property: Change-of-Base for Logarithms
log a x
logb x 
in general
log a b
loge x ln x
1
logb x 


 ln x
loge b ln b ln b
Example:
Find an equation for the derivative and the
value for the derivative at the given x-value.
f ( x)  log7 x
@x 3
1 1
1
f '( x) 
 
ln 7 x x ln 7
1
f '(3) 
 0.171
3ln 7
1
f ( x) 
 ln x
ln 7