Unit 5: Logarithmic Functions Inverse of exponential functions. “log base 2 of 6” Ex 1: Domain: all real numbers Range: y > 0 “log base b of x” Domain: x > 0 Range: all real numbers “base b to x power” “log base 7 of 12” Ex 2:

Logarithmic and Exponential Conversions Convert each log expression into an exponential expression Convert each exponential expression into a log expression (1) Base is always the base (2) Exponent and Answer switch

Example 1CONVERSION PRACTICE a)__________________ b)__________________ c) ______________ d) ______________ ExponentialLogarithmic f)__________________ f) ______________

Example 1: Continued (Fill In The Blanks) a)__________________ c)__________________ b) ______________ d) ______________ ExponentialLogarithmic e)__________________ f) ______________

Useful Log Properties: MEMORIZE THEM!!! Exponential Reasoning [1] [2] [3] [4] Cannot take logs of negative number [5] “ change of base formula” (for calculator)

OPERATION PROPERTIES OF LOGARITHMS #1) Product Property: #2) Quotient Property: #3) Power Property: Log of a product is equal to the SUM of the logs of both multipliers of the same base Log of a quotient “fraction” is equal to the DIFFERENCE of the logs of the numerator and denominator Log of a power statement is equal to the MULTIPLICATION of the power (p) times the log of the power’s base (m)

Useful Log Properties: Examples [1] [2] [3] [5]

OPERATION PROPERTIES OF LOGARITHMS CondenseExpand (1a) (2a) (1b) (3a) (2b) (3b)

Expand Each Logarithm Using Properties (1) (3) (5) (2) (6) (4)

Expand Each Logarithm Using Properties (7) (8) (9) (10) (11)

Condense Each Logarithm Using Properties (1) (4) (5) (2) (6) (3)

(7) (10) (11) (8) (12) (9)

Evaluating Log Expressions: General Rules 1)Set the log expression equal to x 2)Convert log to exponential form 3)Solve the resulting exponential equation for x. “2 raised to what power equals 8?”

Example 2Evaluate using properties a) c) e) b) d) f)

Example 2Evaluate g) i) k) h) j)

Solve Exponential Equations with Logs Solve the exponential until form, b x = a. Clearing Bases Using Log Conversion Some answers cannot be evaluated by hand and require calculator a) b)

a) b) c) d) Solving LOG Equations and Inequalities **SIMPLIFY all LOG Expressions** CASE #1: LOG on one side and VALUE on other Side Apply Exponential Conversion Solve (For inequalities x < # requires 0 < x < # because of domain

a) c) Solving LOG Equations and Inequalities **Simplify all LOG Expressions** CASE #2: LOG on BOTH Bases of both sides should be the same Set the insides of logs equal and Solve b)

Practice: Solving Log Equations – by hand

Practice: Solving Log Equations w/ CALC

Log Property Practice Condense each Log Expression

Use the given values and log properties to evaluate

APPLYING LOG PROPERTIES: SOLVING with PRODUCT PROPERTY [a] [b]

[d] [c]

[a] APPLYING LOG PROPERTIES: SOLVING with QUOTIENT PROPERTY [b]

APPLYING LOG PROPERTIES: SOLVING with POWER PROPERTY [b] [a]

[b] GENERAL PRACTICE [a] [c][d]

GENERAL PRACTICE: Continued [e] [f]