MATHS SYMBOLS - #6 - LOGARITHMS, LOG of a POWER, LOG of a ROOT - PROOFS
0 likes245 views
The document elaborates on the properties and definitions of exponentials and logarithms, providing proofs and examples for various logarithmic identities. It covers definitions, properties, and operations involving logarithms and exponentials in detail, such as the logarithm of a power, root, product, and quotient. The content is organized into sections addressing fundamental concepts and their mathematical proofs.
![logb(1) = - logb(y)
y
OR
- logb(y) = logb(1)
y
Remembering that
logb(yz) = z . logb(y) [log of a Power, previous page]
we get
logb(1) = logb(y-1)
y
= (-1) . logb(y)
= - logb(y)
Q.E.D.
Enzo Exposyto 70](https://image.slidesharecdn.com/maths-symbolslogarithmspowerroot-200712182007/85/MATHS-SYMBOLS-6-LOGARITHMS-LOG-of-a-POWER-LOG-of-a-ROOT-PROOFS-15-320.jpg)
![logb(n√y) = logb(y)
n
Remembering that
logb(yz) = z . logb(y) [log of a Power, page 69]
we get
logb(n√y) = logb(y1/n)
= 1 logb(y)
n
= logb(y)
n
Q.E.D.
Enzo Exposyto 72](https://image.slidesharecdn.com/maths-symbolslogarithmspowerroot-200712182007/85/MATHS-SYMBOLS-6-LOGARITHMS-LOG-of-a-POWER-LOG-of-a-ROOT-PROOFS-17-320.jpg)
![logb(n√yz) = z . logb(y)
n
Remembering that
logb(yz) = z . logb(y) [log of a Power, page 69]
we get
logb(n√yz) = logb(yz/n)
= z logb(y)
n
= z . logb(y)
n
Q.E.D.
Enzo Exposyto 73](https://image.slidesharecdn.com/maths-symbolslogarithmspowerroot-200712182007/85/MATHS-SYMBOLS-6-LOGARITHMS-LOG-of-a-POWER-LOG-of-a-ROOT-PROOFS-18-320.jpg)
MATHS SYMBOLS - #6 - LOGARITHMS, LOG of a POWER, LOG of a ROOT - PROOFS
- 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
- 6. logb(bx) = x PROOFS Enzo Exposyto 61
- 7. logb(bx) = x a) The log of bx is the exponent which we have to put on the base b to get bx itself and, therefore, it’s “x” b) If, from x, we ‘go’ to bx and, then, from bx, we ‘go’ to logb(bx), since logb(bx) is the inverse operation of bx, we go back to x … therefore, logb(bx) = x In other words (remembering that bx = antilogb(x)): logb(antilogb(x)) = x Enzo Exposyto 62
- 8. logb(bx) = x c) Let's set bx = y and, then, we ‘do’ the logarithms in base b of both sides; we get logb(bx) = logb(y) Since bx = y <=> logb(y) = x we can write logb(bx) = logb(y) = x Therefore, logb(bx) = x Q.E.D. Enzo Exposyto 63
- 9. blogb(y) = y PROOFS Enzo Exposyto 64
- 10. blogb(y) = y a) If from y, we ‘go’ to logb(y) and, then, from logb(y), we ‘go’ to blogb(y), since blogb(y) is the inverse operation of logb(y), we go back to y … therefore, blogb(y) = y In other words (remembering that blogb(y) = antilogb(logb(y))): antilogb(logb(y)) = y Enzo Exposyto 65
- 11. blogb(y) = y b) Let's set bx = y and, then, we ‘do’ the log in base b of both sides; we get logb(bx) = logb(y) Remembering that (pages 62-63) logb(bx) = x we can write logb(bx) = logb(y) x = logb(y) or logb(y) = x Now, we ‘do’ the exponentials in base b of both sides and we get blogb(y) = bx Since bx = y we get blogb(y) = y Q.E.D. Enzo Exposyto 66
- 12. blogb(y) = y c) Let's set logb(y) = x and, then, on the left hand side of the equation, we get blogb(y) = bx Since logb(y) = x <=> bx = y it's blogb(y) = bx = y and, therefore, blogb(y) = y Q.E.D. Enzo Exposyto 67
- 13. log of a POWER PROOFS Enzo Exposyto 68
- 14. logb(yz) = z . logb(y) 1) Let's set logb(y) = l and, then, the right hand side of the equation becomes z . logb(y) = z . l 2) Since logb(y) = l <=> bl = y or y = bl the left hand side of the equation becomes logb(yz) = logb((bl)z) = logb(blz) Remembering that (pages 62-63) logb(bx) = x it's logb(blz) = lz = z . l and we can write logb(yz) = logb(blz) = z . l 3) Since the left hand side and the right hand side are equal to z . l, they are equal: logb(yz) = z . logb(y) Q.E.D. Enzo Exposyto 69
- 15. logb(1) = - logb(y) y OR - logb(y) = logb(1) y Remembering that logb(yz) = z . logb(y) [log of a Power, previous page] we get logb(1) = logb(y-1) y = (-1) . logb(y) = - logb(y) Q.E.D. Enzo Exposyto 70
- 16. log of a ROOT PROOFS Enzo Exposyto 71
- 17. logb(n√y) = logb(y) n Remembering that logb(yz) = z . logb(y) [log of a Power, page 69] we get logb(n√y) = logb(y1/n) = 1 logb(y) n = logb(y) n Q.E.D. Enzo Exposyto 72
- 18. logb(n√yz) = z . logb(y) n Remembering that logb(yz) = z . logb(y) [log of a Power, page 69] we get logb(n√yz) = logb(yz/n) = z logb(y) n = z . logb(y) n Q.E.D. Enzo Exposyto 73