Logarithm Bases Coursework

ExponentAlgebraRulesSummaryxnxm... =xn+mMultiplyby same variable - Add exponentsxn/ xm... =xn-mDivideby same variable - Subtractexponents(xn)m... =xnmRaisevariable to a power - Multiplyexponents1/xn... =x-nReciprocalof variable - Change sign of the exponentx0... =1Raisevariable to zero power - Equals 1Logarithm Algebra:The logarithmof a number is the powerto which a basemust be raised to get the number.n = basepowerlogarithmn =power....Example:....1000 = 103For Base 10:logarithm 1000 = 3Any numbercan be expressed as a base raised to a power:....Example:....5.2 = 100.716For Base 10:logarithm 5.2 = 0.716Two types of logarithms are used in general chemistry:...1.Common logarithm, abbreviated log, whose base is 10.2. Naturallogarithm, abbreviated ln, whose base is the irrational number e = 2.71828....Common LogarithmNatural Logarithmlog 10 = 1ln e = 1For: x = 10nn = log x...For: x = emm = ln x...Note that log x is not equal toln x.Rules forcommonandnaturallogs...Summaryln (ab)=ln a + ln b...Multiplyvariables - Addlogarithmsln (a/b)=ln a - ln bDividevariables - Subtractlogarithmsln an=n ln aRaisevariable to a power - Multiplylog by the exponent.eln a=aTo Clear a natural log - Raiselog expression to base e.10log a=aTo Clear a common log - Raiselog expression to base 10.The pKaof a solution is defined by the equation: ... pKa= - log Ka...

Unformatted text preview: Change base: log a b = log c b log c a . Example 75 Differentiate ln( x 2 + 1) . Logarithmic differentiation Example 76 Differentiate y = x 2 + x +5 ( x +1) 2 , x x , (sin x ) x . Number e e = lim x (1 + x ) 1 /x = lim x (1 + 1 x ) x . 30 3.9 Derivative of inverse trigonometric functions Derivative of inverse trig functions: d arcsin x dx = 1 1 x 2 , d arccos x dx = 1 1 x 2 , d arctan x dx = 1 1 + x 2 . Example 77 y = sin(arctan 2 x ) , y = arcsin ( b + a cos x a + b cos x ) . 3.11 Linearization and differentials LINEAR APPROXIMATIONS: we use the tangent line at ( a,f ( a )) as an approximation to the curve y = f ( x ) when x is near a . Definition 14 The approximation f ( x ) f ( a ) + f ( a )( x a ) is called the linear approximation or tangent line approximation of f at a. L ( x ) = f ( a ) + f ( a )( x a ) is called the linearization of f at a. Example 78 a) Find the linearization of the function f ( x ) = x at a = 9 and use it to approximate the numbers 9 . 01 . Sol: x 3 + 1 6 ( x 9) = x 6 + 3 2 , 9 . 01 3 + . 01 6 . b) Are these approximations overestimates or underestimates? Sol: over. c) For what values of x is the linear approximation in a) accurate to within 0.1? Example 79 The linearization of the function f ( x ) = sin x at a = 0 is L(x) = x. Definition 15 If y = f ( x ) , where f is a differentiable function, then the differential dx is an independent variable. That is, dx can be given the value of any real number. The differential dy is then defined in terms of dx by the equation dy = f ( x ) dx . 31 Remark. dy represents the amount that the tangent line rises or falls (the change in the linearization). y represents the amount that the curve y = f ( x ) rises or falls when changes by an amount dx. Example 80 Compare the values of y and dy if y = f ( x ) = x 3 + x 2 2 x + 1 and x changes from: 2 to 2.01. Sol. We have: f (2) = 2 3 + 2 2 2(2) + 1 = 9 , f (2 . 01) = (2 . 01) 3 + (2 . 01) 2 2(2 . 01) + 1 = 9 . 140701 , y = f (2 . 01) f (2) = 0 . 140701 , In general, dy = f ( x ) dx = (3 x 2 + 2 x 2) dx. When dx = x = 0 . 01 , dy = [3(2) 2 + 2(2) 2]0 . 01 = 0 . 14 . Remark. In the notation of differentials, the linear approximation can be written as: f ( a + dx ) f ( a ) + dy. Example 81 The radius of a sphere was measured to be 21 cm with a possible error of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere? Sol. This can be approximated by the differential dV = 4 r 2 dr. When r = 21 and dr = 0.05, this becomes: dV = 4 (21) 2 . 05 277 . 32 Chapter 4. Applications of Derivatives 4.1 Extreme values of functions Absolute (Global) Maximum and Minimum: f ( x ) has a Global (Absolute) Maximum at p if f ( p ) f ( x ) for all x in the domain; f ( x ) has a Global (Absolute) Minimum at p if f ( p ) f ( x ) for all x in the domain; Local (or relative) extreme:...
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