MATH 1004 Lecture Notes - Lecture 14: Natural Logarithm, Binary Logarithm, Logarithm

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$MATH 1004 Full Course Notes$

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Math1004: 4. 1 exponential and logarithms its inverse is. Note: in general: f (x) = ax a = base, a > 0 x = power or exponent. F (x) = loga x x = f (f (x)) = aloga x (cid:3) = aloga (cid:3) or or x = f (f (x)) = loga ax (cid:3) = loga a (cid:3) Example: solve log2 x = 3, evaluate log3. 3: simplify f (x) = log2, write as power function log 1, solve for x log3( x x + 1 x sin x) 0 < x < 3 f (x) = log4(4x8x2+1), x 0 loga(2x2 x) 2(x2 1) = 0: 4. 2 de nition of euler"s number e = 2. 718281 e = lim n (1 + )n = ea: 4. 3 euler"s exponential function and natural logarithm. Properties of ex ex+y = exey ex = 1 + x + ex y = ex/ey x ex = , lim x2.

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Let P denote the vector space of all polynomials and f'(x)denote the derivative of all numbers in P. State true or false tothe following:

1. T: R^2 --> R^3 defined by T(x,y)=(?^2x,0, y/2) is a linear transformation

2. T: R^2 --> R^3 defined by T(x,y)=(x+y, x-y, xy) is alinear transformation

3. T: R^2 --> R^3 defined by T(x,y)=(1,0,0) is a lineartransformation

4. T: R^2 --> R^3 defined by T(x,y)= (y, x, y) is a lineartransformation

5. T: R^2 --> R^3 defined by T(x,y)= (x, x^2, x^3) is alinear transformation

6. T: R^2 --> R^3 defined by T(x,y)= (2x+3y, 3x+4y, 4x+5y) isa linear transformation

7. T: P --> P defined by T(f(x))= f(0) +f'(0)x + f''(0)x^2 isa linear transformation

8. T: P --> P defined by T(f(x))= f(1) + f'(2) + f''(3)x^2 isa linear transformation

9. T: P --> P defined by T(f(x))= f(x-3) is alinear transformation

10. T: P --> P defined by T(f(x))= f(x)-3 is a lineartransformation

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MATH 1004 Lecture Notes - Lecture 14: Natural Logarithm, Binary Logarithm, Logarithm

MATH 1004 Full Course Notes

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