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Logarithm Mid-Unit Assignment

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Transcript of Logarithm Mid-Unit Assignment

Mid-Unit Assignment: History + Intro to Log
David Dobrowolski
MH4U

The Exponential Function and its Inverse
Recall:
f(x) = a(b)
x
initial amount
(y-intercept)
base factor
exponent (variable)
Graphing the function: y=2 results in:
x
D={x E R}
R={y E R/y>0}
Horizontal Asymptote: y=0
Graphing its inverse (x=2 )...
y
D={x E R/x>0}
R={y E R}
Vertical Asymptote: x=0
Determining the equation for f (x)
-1
y=2
x=2
2= x
x
y
y
uh-oh...now what?
History of Logarithms
Napier devoted most of his leisure to the study of mathematics {2}

He began working on logarithms probably as early as 1594 {2}

Logarithms were meant to simplify calculations {1}, especially multiplication for astronomy {2}

The basis for these types of computations were Geometric sequences (Ex: 2, 4, 8, 16 . . . etc.) {2}

Discovered and primarily expressed logarithms in the context of trigonometry so it would be even more relevant to fellow mathematicians through the use of the
"Napier's Bones"
{1}
Attempting to explain this next bit will be kinda hard...
...somehow the Napier Bones...
...leads to trigonometric relationships...
...or also known as...
...the relationship between the 2 lines expressed visually...
...Napier's calculated logarithms from the 2 lines...
...and voila! The precursor for the first ever logarithm table
1. Clark, K. (2011). Logarithms: The Early History of a Familiar Function - John Napier Introduces Logarithms. Retrieved August 19, 2015, from http://www.maa.org/publications/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms

2. Scott, J. (2006, November 24). John Napier. Retrieved August 19, 2015, from http://www.britannica.com/biography/John-Napier

In general: the function y=b can be written in logarithmic form as x=log y, b > 0, b = 1, y > 0
x
b
Ex.1: log 125=x
5
125=5
is in logarithmic form, that can be rewritten as:
x
x=3
Ex.2: log 36=x
6
36=6
x
x=2
Ex.3: log 27=-3
1
_
3
1
_
3
27=
x
( )
x=-3
Expressing Logarithms as Exponentials
Expressing Exponentials as Logarithms
Ex.1 y=10
x=log y
x
logs with no base is really just log x

10
Ex.2 7 =343
x
x=log 343
7
Basic Laws of Logarithms:
log a =x
a
x
a =x
log 1=0
a
log x
a
more complex laws to come next lesson!
In conclusion...
the logarithmic function is the inverse of the exponential function

the function y=b can be written in logarithmic form as x=log y

log x=log x

Basic Laws of Logarithm:
x
b
10
log a =x
a
x
log 1=0
a =x
log x
a
a
References
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