Content:

1. LOGARITHMIC FUNCTIONSPresented by:AMEENA AMEENMARYAM BAQIRFATIMA EL MANNAIKHOLOODREEM IBRAHIMMARIAM OSAMA

2. Content: • Definition of logarithm • How to write a Logarithmic form as an Exponantional form • Properties of logarithm • Laws of logarithm • Changing the base of log • Common logarithm

3. Binary logarithm • Logarithmic Equation • The natural logarithm • Proof that d/dx ln(x) =1/x • Graphing logarithmic functions.

4. AMEENA

5. Definition of Logarithmic Function The power to which a base must be raised to yield a given number e.g. the logarithm to the base 3 of 9, or log3 9, is 2, because 32 = 9

6. The general form of logarithm: • The exponential equation could be written in terms of a logarithmic equation as this form • a^y= Х Loga x = y

7. Example of logarithms:

8. Common logarithms: • The two most common logarithms are called (common logarithms) • and( natural logarithms).Common logarithms have a base of 10 • log x = log10x • , and natural logarithms have a base of e. • ln x =logex

9. Exponential form:- 3^3=27 2^-5=1/32 4^0=1

10. MARYAM.B

11. Properties of Logarithm • because • because • because

12. Property1: loga1=0 because a0=1 • Examples: • (a) 90=1 • (b) log91=0

13. Property 2: logaa=1 because a1=a • Examples: • (a) 21=2 • (b)log22=1

14. Property 3:logaax=x because ax=ax • Examples: • (a) 24=24 • (b) log224=4 • (c) 32=9  log39=2log332=2

15. Property4:blogbx=x • Example: • 3log35=5

16. FATIMA

17. There are three laws of logarithms: Logarithm of products 1 Logarithm of quotient 2 Logarithm of a power 3

18. Remember these laws: The log of 1 is always equal to 0 but the log of a number which is similar to the base of log is always equal 1 1 2

19. Example: Transform the addition into multiplication

20. Example2: Transforming the subtraction into division

21. The form of Will be changed into And the same for Will be Example3:

22. Solve it yourself!

23. KHOLOOD

24. Changing the base: Let a, b, and x be positive real numbers such that and (remember x must be greater than 0). Then can be converted to the base b by the formula let Take the base-c logarithm of each side Power rule Divide each side by

25. * If a and b are positive numbers not equal to 1 and M is positive, then

26. * If the new base is 10 or e, then:

27. Common logarithm: In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm,[ ] . Examples:

28. Binary logarithm: The binary logarithm is the logarithm for base 2. It is the inverse function of . Examples:

29. Binary logarithm: In mathematics, the binary logarithm is the logarithm for base 2. It is the inverse function of . Examples:

30. REEM

31. The Nature of Logarithm Is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.

32. The Nature of Logarithm The natural logarithm can be defined for all positive real numbers x as the area under the curve y = 1/t from 1 to x, and can also be defined for non-zero complex numbers.

33. The Nature of Logarithm The natural logarithm function can also be defined as the inverse function of the exponential function, leading to the identities:

34. Logarithm Equation Logarithmic equations contain logarithmic expressions and constants. When one side of the equation contains a single logarithm and the other side contains a constant, the equation can be solved by rewriting the equation as an equivalent exponential equation using the definition of logarithm.

35. For Example

37. Proof thatd/dx ln(x) = 1/x The natural log of x does not equal 1/x, however the derivative of ln(x) does: The derivative of log(x) is given as:d/dx[ log-a(x) ] = 1 / (x * ln(a))where "log-a" is the logarithm of base a. However, when a = e (natural exponent), then log-a(x) becomes ln(x) and ln(e) = 1: d/dx[ log-e(x) ] = 1 / (x * ln(e)) d/dx[ ln(x) ] = 1 / (x * ln(e)) d/dx[ ln(x) ] = 1 / (x * 1) d/dx[ ln(x) ] = 1 / x

38. MARIAMQAROOT

39. Graphing logarithms is a piece of cake!! • Basics of graphing logarithm • Comparing between logarithm and exponential graphs • Special cases of graphing logarithm • The logarithm families.

40. Graphing Basics: • The important key about graphing in general, is to stick in your mind the bases for this graph. • For logarithm the origin of its graph is square-root graph..

41. (b,1) 1 1 b Before graphing y= logb (x) we can start first with knowing the following: The logarithm of 1 is zero (x=1), so the x-intercept is always 1, no matter what base of log was. For example if we have: b = 2 power 0 = 1 b = 3 power 0 = 1 b = 4 power 0 = 1 Values of x between 0 and 1 represent the graph below the x-axis when: Fractions are the values of the negative powers.

42. Examples on graphing logarithm: • EXAMPLEONE Graph y = log2(x). First change log to exponent form: X=2 power y, then start with a T-chart

43. EXAMPLEtwo: Graph First change ln into logarithm form: Loge (x) Then change to exponential form: X= e power y..Now draw you T-chart

44. EXAMPLEtwo: Graph y = log2(x + 3). This is similar to the graph of log2(x), but is shifted "+ 3" is not outside of the log, the shift is not up or down First plot (1,0), test the shifting The log will be 0 when the argument, x + 3, is equal to 1. When x = –2. (1, 0) the basic point is shifted to (–2, 0) So, the graph is shifted three units to the left draw the asymptote on the x= -3

45. The graph of y = log2(x + 3) Looks like this:

46. Remember: • You may get some question about log like for example: • Log2 (x+15) = 2 • Solution: • 2^2= x+15 • x= -11, which can never be real • Therefore, No Solution

47. Compare between logarithm and exponential graphs:

48. The Equations  y = b x  and  x = log b y  say the same thing.

49. y =- loga x y = loga x

50. Y=loga(x+2) y = log2 (-x) .