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Uniqueness Theorem and Properties of Log Functions

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. Log of a Product.

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Uniqueness Theorem and Properties of Log Functions

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  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. 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.

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