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1. Logarithm

2. Logarithm (Introduction) The logarithmic function is defined as the inverse of the exponential function. *A LOGARITHM is an exponent.  It is the exponent to which the base must be raised to produce a given number. For b > 0 and b 1 is equivalent to Number Exponent Base

3. Examples, • Since , then • Since , then • Since , then • Since , then

4. Properties of Logarithm • because • because • because

5. Rules of Logarithm 1. 2. 3.

6. Changing the base: IF: ,Then Y= Now we solve For y, using base-b logarithms: If: Take the base-b logarithm of each side Power rule Divide each side by

7. Base-change formula: If a and b are positive numbers not equal to 1 and M is positive, then

8. If the new base is 10 or e, then

9. Example :find To four decimal places • Solution: By using the base-change with a=7 and b=10: Chick by finding with calculator. Note that we also have

10. Common logarithms • common logarithm is the logarithm with base 10. • It is indicated by or sometimes Log(x) with a capital L • Traditionally, log10 is abbreviated to log.

11. Example :

12. Binary logarithm • In mathematics, the binary logarithm is the logarithm for base 2. It is the inverse function of . • Domain and range: the domain of the exponential function is and its range is Because the logarithm function is the inverse of The domain of is and its range is

13. Examples :

14. Logarithmic equation : If we have equality of two logarithms with the same base, we use the one-to-one property to eliminate the logarithm. If we have an equation with only one logarithm, such we use the definition of logarithm to write and to eliminate the logarithm

15. Find the solution : • Solution: Original equation Take log of each side Power rule Distributive property Get all x-terms on one side Factor out x Exact solution

16. Example (2) Solve Solution : Original equation Product rule Multiply the binomials Definition of logarithm Even root property

17. To check, first let x=-5 in the original equation : Because the domain of any logarithmic function is the set of positive numbers, these logarithms are undefined. Now check x=5 in the originalequation: The solution is {5}. Incorrect Correct

18. Natural Logarithm • The natural logarithm is a logarithm to base e • Where e = 2.7182818…. • it is denoted ln x, as ln x = loge x

19. Reason for being "natural" The reason we call the ln(x) "natural" : • expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10 • the natural logarithm can be defined quite easily using a simple integral or Taylor series--which is not true of other logarithms • there are a number of simple series involving the natural logarithm, and it often arises in nature. Nicholas Mercator first described them as log naturalis before calculus was even conceived.

20. The general definition of a logarithm Y = ln x means the same as x = ey And this leads us directly to the following: • ln 1 = 0 because e0= 1 • ln e = 1 because e1= e • ln e2= 2 and ln e-3= -3

21. Properties: • All the usual properties of logarithms hold for the natural logarithm, for example: • (where 28 is an arbitrary real number) • ln (x)a = a ln x

22. Example 1: • ln(e)4= 4ln(e) = 4(1) & since ln e = 1

23. Example 2:

24. Example 3: • ln 5e= = ln 5 + ln e = ln ( 5 )+ 1

25. Example 4: • It doesn't exist! Why?

26. Example 5 • e ln 6 = ? • e ln 6 = 6

27. Proof that d/dx ln(x) = 1/x • F (x) = ln(x) • . f ‘ (x) = lim h-->0 (f (x + h) – f (x)) /h • Definition of a derivative • = lim h-->0 (ln(x + h) - ln(x))/h • Plugging the function f (x) = ln(x) • = lim h-->0 ln( (x + h) /x) /h • Rule of logarithms: log (a) – log (b) = log (a/b)

28. = lim h-->0 ln(1 + h/x)/h • Algebraic simplification: (x + h)/x = 1 + x/h • = lim h-->0 ln(1 + h/x)⋅(x/h)⋅(1/x) • Algebraically, 1/h = (x/h)(1/x) • = 1/x ⋅ lim h-->0 ln(1 + h/x) ⋅ (x/h) • 1/x is a constant with respect to the variable being "limited," so we can pull it out of the limit .

29. = 1/x ⋅ lim_h-->0 ln((1 + h/x)x/h) • Rule of logs: log(a) ⋅ b = log(ab) • Let's look at a definition of e using a limit: • e = lim n-->∞ (1 + 1/n)n Or equivalently: e = lim n-->0 (1 + n)1/n • lim h-->0 (1 + h/x) x/h = e • True from the definition of e (the x is irrelevant, since it's constant with respect to h) • 1/x ⋅ lim_h-->0 ln((1 + h/x)x/h)

30. = 1/x ⋅ ln(e) • Follows from (7.5) applied to (7)Since e is the base of ln: ln (e) = 1 • = 1/x • What happens when you multiply anything by 1 is that it doesn't change.

31. Compare between the graphs of:

32. (b,1) 1 1 b • for any base ,x-intercept is 1. • because the logarithm of 1 is 0 . 2)The graph passes through the point (b,1) . because the logarithm of the base is 1. 3) The graph is below the x-axis, the logarithm is negative for Which number are those that have negative logarithms.

33. Ex:(4)Graph the following Sol: Change to exponential form,

34. Ex:(5)Graph the following • Sol: • Change to exponential form,

35. Sol: Change to exponential form, Ex:(3)Graph the following.

36. Compare between the graphs of:

37. EX: Solve S0l: No solution

38. Thank You