**Mid-Unit Assignment: History + Intro to Log**

**David Dobrowolski**

MH4U

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