Pre-Cal 3.1 Exponential Functions. -Transforming exponential graphs: -natural base e: = 2.71828… -To solve an exponential equation: 1. Rewrite the powers.

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Presentation transcript:

Pre-Cal 3.1 Exponential Functions

-Transforming exponential graphs: -natural base e: = … -To solve an exponential equation: 1. Rewrite the powers so both sides have a common base 2. Set exponents equal and solve for x

Precal 3.2 Logarithmic Functions -Logarithmic Function: a function that can be written in the form: f(x) = log a x Note: f(x) = log a x and f(x) = a x are inverse functions -Converting between exponential and logarithmic form: y = log a x if and only if a y = x

Remember: Logarithms are Exponents!! -Evaluating Logarithms: 1. Set the log equal to x 2. Rewrite in exponential form to solve -Natural Logarithm: -Common Logarithm: -Graphs of Logarithmic Functions: 1. Basic graph: y = log a x

Precal 3.3 Properties of Logarithms -Basic Properties: 1. log a 1 = 0 2.log a a = 1 3. log a a x = x -More Properties: 4. Product Rule: 5. Quotient Rule: 6. Power Rule:

-Change of Base Formula: (if we use a = e) (if we use a = 10)

Precal 3.4 Logarithmic Equations -To solve a logarithmic equation: 1. Isolate logarithm on one side 2. Rewrite in exponential form 3. Solve for the variable 4. For base e equations, take the ln of both sides **calculator**

Precal 3.5 Exponential Growth & Decay -Given an exponential function: 1. If a > 0 and k > 0, f(x) is a growth function 2. If a > 0 and k < 0, f(x) is a decay function -Growth & Decay Models: -Exponential Population Model: P 0 = initial pop. r = percentage rate change t = time

Precal 3.6 Basic Combinations and Permutations -Multiplication Principle of Counting: If a procedure P has a sequence of S stages that can occur in R ways, then the number of ways that the procedure P can occur is the product of the R ways S stages can occur -Combinations: the unordered selection of objects from a set. - Permutations : the ways that a set of n objects can be arranged in order.

-The number of combinations of n objects taken r at a time is: -The number of permutations of n objects taken r at a time is: - n factorial = n! = special case: 0! = 1

Precal 3.7 Expanding Binomials -The symbol: -The formula: The Binomial Theorem:

-Pascal’s Triangle: