Logarithms and Logarithmic Functions

- Coach Baughman
- November 20, 2003
- Algebra II

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Objectives

- The students will identify a logarithmic

function. (Knowledge) (Mathematics, Algebra II,

6.a) - The students will solve logarithmic expressions.

(Application) (Mathematics, Algebra II, 6.b) - The students will solve logarithmic functions.

(Application) (Mathematics, Algebra II, 6.c)

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John Napier

- Born in Edinburgh, Scotland, in 1550
- Began education at St. Andrews University at the

age of 13 - Likely acquired mathematical knowledge at the

University of Paris - Died April 4, 1617 in Edinburgh, Scotland

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Logarithms

- Definition If b and y are positive where b?1,

then the logarithm of y with base b (logby) is

defined as - logby x
- if and only if bx y.

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Special Logarithms

- logb1 0
- Why? b0 1
- logbb 1
- Why? b1 b
- The logarithm with base 10 is called the common

logarithm. (log10 or log) - The logarithm with base e is called the natrual

logarithm. (loge or ln)

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Examples

- Evaluate the expression log381

- 3x 81

- 3x 34

- x 4

- Evaluate the expression log1/28

- (1/2)x 8

- (1/2)x 23

- x -3

- (1/2)x (1/2)-3

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Logarithmic Functions

- Exponential functions and logarithmic functions

are inverses - undo each other
- If g(x) logbx and f(x) bx, then g(f(x))

logbbx x and - f(g(x)) blogbx x.

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Examples

- Simplify the expression 10log2

- 10log2 2

- Simplify the expression log39x

- log39x log3(32)x

- log332x

- 2x

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More Examples

- Find the inverse of y log3x

- Use the definition of a logarithm

- y 3x

- Find the inverse of y ln(x 1)

- y ln(x 1)

- x ln(y 1) (switch x and y)

- ex y 1 (write in exponential form)

- ex 1 y (solve for y)

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Assessment

- Write log7b 13 in exponential form.
- Write 43 64 in logarithmic form.
- Solve the equation logx(1/32) -5.
- Simplify log5252.
- Evaluate log4256.
- Find the inverse of y ln(2x 5)

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Closing Questions

- What did we learn about today?
- Can anyone tell me the definition of a logarithm?
- Where might you use logarithms?

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