Uniqueness Theorem and Properties of Log Functions

Lesson 6-3Logarithm 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:

c d c db b b

cc d

d

bb

b

dc c db b

log ( ) log logb b bcd c d

log log logb b bc

c dd

log ( ) logrb bc r c

Product of powers

Quotient of powers

Power of a power

Log of a Product

Log of a Quotient

Log of a Power

Where b>0, b≠1,c>0,d>0 and r is any real #

Algebraic Definition of Logarithm

log if and only if aba c b c

10log (100000)a

10 100000a 5Since 10 100000

5a

- if functions start at the same point and change the same way, they are the same.

- this relies on the Mean Value Theorem

Uniqueness Theorem for Derivatives

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:

ln( ) lnra r a

ln( ) ln lnab a b

ln ln lna

a bb

Quotient:

Product:

Power:

Intercept:1

1

1ln(1) 0 since 0dt

t

Examples:

ln(21) ln 7 ln3

3.045 1.946 1.099

3.045 3.045

Evaluate both sides of the equations to show they are equivalent.

28ln ln 28 ln 4

4

1.946 3.332 1.386

1.946 1.946

Examples:

3ln(6 ) 3ln 6

5.375 3(1.792)

5.375 5.376

Evaluate both sides of the equations to show they are equivalent. ln(1) 0

Examples:

1 @ 2.7182818y x

ln( ) 1x

For what value of x is ln equal to 1?

x e

Log in bases other than 10.

Property: Equivalence of Natural Logs and Base e Logs

Property: Change-of-Base for Logarithmsln log for all x>0ex x

loglog in general

loga

ba

xx

b

log ln 1log ln

log ln lne

be

x xx x

b b b

Example:

Find an equation for the derivative and the value for the derivative at the given x-value.

7( ) log @ 3f x x x 1

( ) lnln 7

f x x 1 1 1

'( )ln 7 ln 7

f xx x

1'(3) 0.171

3ln 7f