1. 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.

2. Exponential and Logarithmic Properties Correspond:
Product of powers
Log of a Product
Quotient of powers
Log of a Quotient
Log of a Power
Power of a power
Where b>0, b≠1,c>0,d>0 and r is any real #

3. Algebraic Definition of Logarithm

4. 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.

5. Logarithm Properties of Ln:
Product:
Quotient:
Power:
Intercept:

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

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

8. Examples:
For what value of x is ln equal to 1?

9. Log in bases other than 10. and
Property: Equivalence of Natural Logs and Base e Logs
Property: Change-of-Base for Logarithms
and

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