PRECALCULUS NYOS CHARTER SCHOOL QUARTER 4 “IF WE DID ALL THE THINGS WE WERE CAPABLE OF DOING, WE WOULD LITERALLY ASTOUND OURSELVES.” ~ THOMAS EDISON Logarithmic.

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Presentation transcript:

PRECALCULUS NYOS CHARTER SCHOOL QUARTER 4 “IF WE DID ALL THE THINGS WE WERE CAPABLE OF DOING, WE WOULD LITERALLY ASTOUND OURSELVES.” ~ THOMAS EDISON Logarithmic Functions

 The logarithmic function y = log a x, where a > 0 and a ≠ 1, is the inverse of the exponential function y = a x. y = log a x iff x = a y

Logarithmic Functions Example: Write in exponential form. log 3 9 = 2

Logarithmic Functions

Properties of Logarithms PropertyDefinition Productlog b mn = log b m + log b n

Logarithmic Functions Example: Expand log 5 9x = log log 5 x

Logarithmic Functions Example: Expand log x 12y

Logarithmic Functions Example: Expand log x 12y = log x 12 + log x y

Logarithmic Functions Properties of Logarithms PropertyDefinition Productlog b mn = log b m + log b n Quotient

Logarithmic Functions Example: Expand log 5 9/x = log log 5 x

Logarithmic Functions Example: Expand log x 12/y

Logarithmic Functions Example: Expand log x 12/y = log x 12 - log x y

Logarithmic Functions Properties of Logarithms PropertyDefinition Productlog b mn = log b m + log b n Quotient Powerlog b m p = p log b m

Logarithmic Functions Example: Simplify log 5 9 x = x log 5 9

Logarithmic Functions Properties of Logarithms PropertyDefinition Productlog b mn = log b m + log b n Quotient Powerlog b m p = p log b m EqualityIf log b m = log b n, then m = n

Logarithmic Functions Example: Simplify. log 5 9 = log 5 x 9 = x

Logarithmic Functions Example: Solve for x. log 5 16 = log 5 2x 16 = 2x 8 = x

Logarithmic Functions Properties of Logarithms PropertyDefinition Productlog b mn = log b m + log b n Quotient Powerlog b m p = p log b m EqualityIf log b m = log b n, then m = n Identitylog a a = 1 Zerolog a 1 = 0

Logarithmic Functions Example: Simplify. log 5 5 1

Logarithmic Functions Example: Simplify. log

Logarithmic Functions Example: Simplify. log

Logarithmic Functions Example: Simplify. log

Logarithmic Functions Example: Solve. log 8 48 – log 8 w = log 8 6

Logarithmic Functions Example: Solve. log 8 48 – log 8 w = log 8 6 log 8 (48/w) = log /w = 6 w = 8

Logarithmic Functions

Example: Convert log 5 43 to a natural logarithm and evaluate. log 5 43

Logarithmic Functions

Example: Solve. 9 x-4 = 7.13

Logarithmic Functions

Example: Solve. 6 x+2 = 14 The variable is in the exponent. Take the log of both sides. ln 6 x+2 = ln 14

Logarithmic Functions

Example: Solve. 2 x-5 = 11

Logarithmic Functions

Example: Solve. 6 x+2 = 14 x-3

Logarithmic Functions Example: Solve. 6 x+2 = 14 x-3 ln 6 x+2 = ln 14 x-3 Move the exponents to the front and distribute… x ln ln 6 = x ln 14 – 3 ln 14 Get the x terms on the left side and constants on the right… x ln 6 - x ln 14 = – 3 ln 14 – 2 ln 6 Factor out an x from the left side… x (ln 6 - ln 14) = – 3 ln 14 – 2 ln 6

Logarithmic Functions

Example: As a freshman in college, McKayla received $4,000 from her great aunt. She invested the money and would like to buy a car that costs twice that amount when she graduates in four years. If the money is invested in an account that pays 9.5% compounded continuously, will she have enough money for the car?

Logarithmic Functions

Example: What interest rate is required for an amount to double in 4 years?

Logarithmic Functions