1.5 Functions and Logarithms Golden Gate Bridge San Francisco, CA Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington

A relation is a function if: for each x there is one and only one y. A relation is one-to-one if also: for each y there is one and only one x. In other words, a function is one-to-one on domain D if: whenever

To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. not one-to-one not a function (also not one-to-one)

Inverse functions: Given an x value, we can find a y value. Solve for x: Inverse functions are reflections about y = x. Switch x and y: (eff inverse of x)

example 3: Graph: for a parametrically: Y= WINDOW GRAPH

b Find the inverse function: example 3: Graph: for b Find the inverse function: WINDOW Switch x & y: Change the graphing mode to function. Y= > GRAPH

This is a one-to-one function, therefore it has an inverse. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Two raised to what power is 16? Example: The most commonly used bases for logs are 10: and e: is called the natural log function. is called the common log function.

In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work. is called the natural log function. is called the common log function.

A Few Historical Notes: English translation by Edward Wright Logarithms were "invented" by a Scottish nobleman named John Napier (1550-1617). Logarithm is the combination of two Greek roots, Logos (reason or ratio) + artihmus (number). The word logarithm was introduced in Napier’s 1614 work, Mirifici Logarithmorum canonis descriptio, (description of the wonderful canon of logarithms), originally published in Latin.

Astronomer Johannes Kepler read Napier’s work in 1616, and used logarithms in developing his Third Law of Planetary Motion. Kepler’s Third Law states: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Kepler published the third law in 1620 in a book titled Euphemerides, dedicated to Napier. He later published his own work on logarithms.

The term “natural logarithm” (actually “logarithm Ephemerides us naturalis”) was first used by an Italian mathematician, Pietro Mengoli (1626-1686). Surprisingly, the notation ln for natural logs was not first used until 1893 by an American, Washington Irving Stringham (1847-1909). Washington Irving Stringham was a mathematics professor at the University of California in Berkley. He was a graduate of Harvard University and earned his PhD from Johns Hopkins University. He also served temporarily as president of the University of California and had been chair of the mathematics department for 27 years when he died. “In place of elog we shall henceforth use the shorter symbol ln , made up of the initial letters of logarithm and of natural or Napierian”

Even though we will be using natural logs in calculus, you may still need to find logs with other bases occasionally. Here is a useful keyboard shortcut for the newer TI-89 Titanium calculators. (Unfortunately the shortcut does not work on the older TI-89s.) 7 returns: If you enter: you get: (base 10) If you enter: you get: (base 2)

9 And while we are on the topic of TI-89 Titanium keyboard shortcuts: returns: 9 If you enter: you get: (fifth root)

For a quick key to all of the TI-89 Titanium shortcuts, press: EE KEY

Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

p $1000 is invested at 5.25 % interest compounded annually. Example 6: $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent. 17.9 years In real life you would have to wait 18 years. p