Logarithms

Let’s Get It Started!!!

Remember  A logarithm is an exponent  Every time you are working with logarithms, you can substitute the word exponent for logarithm,

The symbol for logarithm is log What is written log 3 9 What is said log to the base 3 of 9 or log base 3 of 9

What does it mean???? log 3 9 means what power of 3 is 9?

log 3 9 = 2 because 3 2 = 9 log 3 9 = 2 because 3 2 = 9 ^ ^ ^ ^ logarithmic form exponential form logarithmic form exponential form

Write in exponential form: Logarithmic Form log 2 8 = 3 log 6 36 = 2 log 4 1 = 0 log 2 32 = 5 Exponential Form 2 3 = = = = 32

Write in logarithmic form Exponential Form 3 0 = = = = 216 Logarithmic Form log 3 1 = 0 log = 4 log 10 (.1) = -1 log = 3

Special log values log a 1 = 0 because log a a = 1 because log a a x = x because a 0 = 1 a 1 = a a x = a x

We interrupt the lesson on logarithms to bring you an important message… Hold on!! Let’s make sure you understand what “if and only if” means.

A triangle is scalene if and only if the triangle has 3 sides of different lengths. The above statement takes the place of 2 statements: A triangle is scalene if it has 3 sides of different lengths. AND A triangle with 3 sides of different lengths is scalene.

Now back to the lesson on logarithms…

Definition The logarithm of x with base a (a > 0) is denoted by log a x and is defined as y = log a x iff a y = x if and only if

y = log a x Replace the y with f (x) and you have a function f ( x ) = log a x

Evaluate: log 5 25 = log 4 64 = log 2 64 = log 4 2 = log 3 (-1) = ½ Impossible Why????

The log function with base 10 is called the common log function Your calculator uses base 10 when doing logarithms.

If using common logs, you do not need to write in the base of 10. log 100 = 2 because 10 2 = 100

If you have any other base, to use your calculator, you can use the CHANGE OF BASE FORMULA: log a x = log x / log a

Use your calculator to evaluate: log 4 7 = log 7 / log 4  1.404

Log 3 40 = log 40 / log 3  3.358

More Examples using the calculator log = log 4 84 = log 2 79 = log 543 / log 3  log 84 / log 4  log 79 / log 2  6.304

Graph f ( x ) = log 2 x Make a table of values xyxy

Let’s review logarithms A logarithm is an exponent Calculators use base 10 To find log 5 399, using a calculator, use log 399 / log 5 (  3.72)

The End!

HOMEWORK TIME!!!