MATHS SYMBOLS - #2 - EXPONENTIALS and LOGARITHMS
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The document discusses the properties and definitions of exponentials and logarithms, using various examples and proofs to illustrate their relationships. It explains how logarithms serve as the inverse operations of exponentials and introduces concepts like antilogarithms. Key principles are demonstrated through base 2 logarithms and the concept of exponentiation.
MATHS SYMBOLS - #2 - EXPONENTIALS and LOGARITHMS
- 1. ENZO EXPOSYTO MATHS SYMBOLS PROPERTIES of EXPONENTIALS and LOGARITHMS Enzo Exposyto 1
- 2. 2X ex 2-x e-x EXPONENTIALS LOGARITHMS log2(x) ln(x) log(x) Enzo Exposyto 2
- 4. 1 - Exponential - Definition 6 2 - Exponentials - Their Properties 8 3 - Exponentials and Logarithms 15 4 - Logarithm - Definition and Examples 25 5 - Logarithms - Their Properties 34 6 - log(y) and ln(y) - Properties 46 Enzo Exposyto 4
- 5. 7 - logb(bx) = x - Proofs 61 8 - ylogb(y) = y - Proofs 64 9 - log of a Power - Proofs 68 10 - log of a Root - Proofs 71 11 - log of a Product - Proofs 74 12 - log of a Quotient - Proofs 77 13 - Change of Base - Proofs 82 14 - zlogb(y) = ylogb(z) - Proof 91 15 - SitoGraphy 93 Enzo Exposyto 5
- 7. EXPONENTIAL BASE 2: 23 = 2 × 2 x 2 = 8 The LOGARITHM BASE 2 OF 8 goes the OTHER WAY: Enzo Exposyto 16
- 8. The logarithm base 2 of 8 is 3, BECAUSE 2 cubed is 8; so the logarithm base 2 of 8 is 3: Enzo Exposyto 17
- 9. THE LOGARITHM BASE 2 OF 8 IS THE OPERATION THAT ALLOWS US OF GOING BACK TO THE EXPONENT 3 EXPONENTIAL 2x and LOGARITHM BASE 2 EXPONENT x 2x 1 2 2 4 3 8 4 16 Enzo Exposyto 18
- 10. In other words … THE LOGARITHM BASE 2 OF 8 IS THE EXPONENT 3 … MORE PRECISELY … THE LOGARITHM "3" IS THE EXPONENT WHICH WE HAVE TO PUT ON THE BASE “2” TO GET “8” Enzo Exposyto 19
- 11. Now, since log2(8) = 3 and 8 = 23 then log2(23) = 3 We can see that THE LOGARITHM "3" IS THE EXPONENT WHICH WE HAVE TO PUT ON THE BASE “2” TO GET “23” Enzo Exposyto 20
- 12. EXPONENTIAL and LOGARITHM with the same base CANCEL EACH OTHER. This is true because exponential and logarithm with the same base are INVERSE OPERATIONS It is just like Addition and Subtraction, Multiplication and Division, Exponentiation and Root, … when they're INVERSE OPERATIONS Enzo Exposyto 21
- 13. Now, we can introduce the ANTILOGARITHM BASE b: antilogb(x) = bx It's, simply, an EXPONENTIAL and represents the antilogarithm when we operate with a logarithm … It’s such that logb(antilogb(x)) = x The meaning is logb(bx) = x Enzo Exposyto 22
- 14. And, of course, antilogb(logb(y)) = y The meaning is blogb(y) = y Enzo Exposyto 23
- 15. These phrases - with 8 - are equivalent log2(8) = 3 < = > 23 = 8 OR 23 = 8 < = > log2(8) = 3 These phrases - with 23 - are equivalent log2(23) = 3 < = > 23 = 23 OR 23 = 23 < = > log2(23) = 3 Enzo Exposyto 24