**Logarithms**

Parts of a logarithm

Base: the number we are multiplying (i.e. 2)

Logarithm: how many times you multiply the base (i.e. 3)

The number we want to get (i.e. 8)

What is a logarithm???

Definition: a quantity representing the power to which a fixed number (the base) must be raised to produce a given number

Simpler form: how many of one

number

do we multiply to get

another number

?

Logarithms & Exponents

The logarithm tells you what the exponent is!

**By: AJ Foremny, Bobby Reilly, & Jen Stief**

Memory Strategy: Mnemonics

Logarithmic and exponential functions are inverses of eachother

Applications: pH (measure acidity), decibels (sound intensity), Richter Scale

Types of Logarithms

Common Log

(Base 10)

Natural Log

(Base e)

Applications: exponential growth & decay models (i.e. compounding), radioctive decay (half-life), population growth

History of Logarithms

References

Logarithms were originally invented by John Napier in 1614; these did not have bases

The purpose was to simplify multiplication and division to the level of addition and subtraction

Napier and Henry Briggs invented the common system of logarithms 1624

Defining logarithms as exponents wasn't recognized until 1685 by John Wallis

Logarithm Tables

Slide Rule

In 1622 English mathematician William Oughtred invented the slide-rule for multiplying and dividing numbers based on logarithms.

Numbers are placed on slide rule scales at distances proportional to the differences between their logarithms. By mechanically adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale, one can quickly determine that 2 x 3 = 6.

Future Technology

MIT engineers have created synthetic biology circuits that can perform analog computations such as taking logarithms and square roots in living cells.

Logarithms Using a Calculator

Scientific and graphing calculators can quickly calculate logarithms

Flaws of the TI-89

There is no log button. In order to calculate with logarithms, you must:

press CATALOG then find log

press APLHA then type in log

press diamond then 7

Logarithms with Different Bases

To enter a logarithm with a different base on the calculator, use the Change of Base Formula:

Example:

When no base is indicated, base 10 is implied. The log key will calculate the common logarithm. 2nd log will calculate the antilogarithm or 10^x.

The ln key calculates the natural logarithm. 2nd ln will calculate e^x.

TI-73 Explorer

There is no log button, but there is a LOG menu!

John Napier