Logarithmic Functions Done by: Jihan Haya Fatma Laila Maryam Moodey

Main points History Why do we care Definition of logarithm Common logarithm Laws of logarithm Changing the base Graphing logarithmic function

History Logarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, a Swiss. The logarithms which they invented differed from each other and from the common and natural logarithms now in use. Napier's logarithms were published in 1614; Burgi's logarithms were published in 1620. The objective of both men was to simplify mathematical calculations. Napier's approach was algebraic and Burgi's approach was geometric. Neither men had a concept of a logarithmic base. Napier defined logarithms as a ratio of two distances in a geometric form. The possibility of defining logarithms as exponents was recognized by John Wallis in 1685 and by Johann Bernoulli in 1694.

History Today the use of the logarithms for calculations is obsolete, but the concept remains fundamental in the basic mathematical culture and they are very present as well in physics as in chemistry. Their history undoubtedly remains a modest chapter, but its exemplarity, and its richness testify to what can present the development of Mathematics.

Why do we use logarithms ? Why do we care ? Why do we use logarithms ? To model many natural processes. To measure the pH or acidity of a chemical solution. To measure earthquake intensity on the Richter scale. To analyze exponential processes.

Logarithmic functions On this project I hope to clear up problems you might have with exponential and/or logarithmic functions.  Logarithms are used all the time in real life, for example, the Richter scale, and are very useful for measuring things that grow or diminish exponentially.  When you look to the explanation you will gain a better understanding of exponential and logarithmic functions.

Inverse functions Find f`(x) of 3x + 1 To Understand the Logarithmic Function you need to know what is: Inverse functions To find the inverse of a function (the inverse of a function is the same as reflecting a function across the line y = x), interchange x and y and then solve for y.  The inverse of f(x) is denoted by f-1(x). Example: Find f`(x) of 3x + 1

Solution x = 3y + 1 x - 1 = 3y (x - 1)/3 = y f-1(x) = (x - 1)/3 The equation is y = 3x + 1. Interchange x and y. x = 3y + 1 Solve for y. x - 1 = 3y (x - 1)/3 = y f-1(x) = (x - 1)/3

Exponential functions Exponential functions are functions where f(x) = ax + B where a is any real constant and B is any expression. Example: f(x) = e-x - 1 is an exponential function.

log x = a if and only if 10a = x Common function log x = a if and only if 10a = x   The important thing to remember is the log represents the exponent.  In the case of common logs, the base is always base 10.  Study the following examples.

1)  log 100 = 2 because 102 = 100. 2)  log 1000 = 3 because 103 = 1000. 3)  log 1 = 0 because 100 = 1. 4)  log .1 = -1 because 10-1 = .1 5)  log .01 = -2 because 10-2 = .01

Definition of Logarithmic Function The logarithmic function with base b, where b > 0 and b =1, is denoted by logb and is defined by  y=logb x if and only if by= x

8 = 2x x = log2 8 Example 1 Convert to logarithmic form: Solution Remember that the logarithm Is the exponent. x = log2 8

Convert to exponential form: y = log3 5 Solution Example 2 Convert to exponential form: y = log3 5 Solution Remember that the logarithm Is the exponent. 3y = 5

The figure below is a little chart that always helped us remember how to convert from exponential to logarithmic form and from logarithmic to exponential form.

Convert the logarithm to exponential Form. Example 3 Solve log2 x = -3 Solution Convert the logarithm to exponential Form. 2-3 = x x = (1/8)

Rules of Logarithmic 1)Logb MN = Logb M + Logb N Example: log(3x) = log (3) + log (x) 2)Logb M/N = Logb M - Logb N Example: Log 2/3 = log (2) – log (3)

   3)  Logb M = Logb N if and only if M = N Example: log (x) = log (5-x), X=5-x.   4)  Logb Mk = k Logb M Log36 = 6log3   

5)Logb bk = k Example: Log 106 = 6 6)Logb b = 1 Log 10 = 1 7)  Log b 1 = 0

When you give, for example; log a x and was told to find the answer using a calculator and in the calculator log in to the base 10, so the only solution is to change the base to ten or in some cases to (e) and there is a formula, that you can use to convert the base of the log to ten, which is: Let a, b, and x be positive real numbers such that (a = 1) and (b) is equal either 10 or (e). In the formula (a) can not be equal to 1 because log 1 is equal 0 and as we know the nominator can not be zero or the answer will be undefined.

Find to an accuracy of six decimals: Example(1): Find to an accuracy of six decimals: Calculate Equal Answer = 1.771244

Checking the Answer or

Example(2): Calculate Equal Answer = 1.771244

Graphing logarithmic function Graph the function y = f(x) = log3 x First, we need to write in exponential form 3y = x

Put in the same values for y each time and then find it’s corresponding x value for the given function. x y x = 3y ( x , y ) -2 x = 3-2 =1/9 ( 1/9 , -2 ) -1 x = 3-1 = 1/3 ( 1/3 , -1 ) x = 30 = 1 ( 1 , 0 ) 1 x = 31 = 3 ( 3 , 1 ) 2 x = 32 = 9 ( 9 , 2 )

Plot the points Draw the curve

1- We need to write in exponential form Example (2): Graph the function 1- We need to write in exponential form 3y = x+ 1 3y – 1 = x

Put in the same values for y each time and then find it’s corresponding x value for the given function. x y x = 3y - 1 ( x , y ) -2 x = 3-2 – 1 = 1/9 – 1 = -8/9 ( -8/9 , -2 ) -1 x = 3-1 – 1 = 1/3 -1 = -2/3 ( -2/3 , -1 ) x = 30 – 1 = 1 – 1 = 0 ( 0 , 0 ) 1 x = 31 – 1 = 3 – 1 = 2 ( 2 , 1 ) 2 x = 32 – 1 = 9 – 1 = 8 ( 8 , 2 )

Plot the points Draw the curve

QUIZ

Questions log4 1/16 Answer = -2 because 4-2 = 1/16

Questions Logb w – logb x – logb y Answer Logb w – ( logb x + logb y ) = logb w / logb xy