Logarithm and Exponential Questions with Answers and Solutions - Grade 12

Grade 12 questions on Logarithm and exponential with answers and solutions are presented.

 Solve the equation (1/2)2x + 1 = 1 Solve x ym = y x3 for m. Given: log8(5) = b. Express log4(10) in terms of b. Simplify without calculator: log6(216) + [ log(42) - log(6) ] / log(49) Simplify without calculator: ((3-1 - 9-1) / 6)1/3 Express (logxa)(logab) as a single logarithm. Find a so that the graph of y = logax passes through the point (e , 2). Find constant A such that log3 x = A log5x for all x > 0. Solve for x the equation log [ log (2 + log2(x + 1)) ] = 0 Solve for x the equation 2 x b4 logbx = 486 Solve for x the equation ln (x - 1) + ln (2x - 1) = 2 ln (x + 1) Find the x intercept of the graph of y = 2 log( sqrt(x - 1) - 2) Solve for x the equation 9x - 3x - 8 = 0 Solve for x the equation 4x - 2 = 3x + 4 If logx(1 / 8) = -3 / 4, than what is x? Solutions to the Above Problems Rewrite equation as (1/2)2x + 1 = (1/2)0 Leads to 2x + 1 = 0 Solve for x: x = -1/2 Divide all terms by x y and rewrite equation as: ym - 1 = x2 Take ln of both sides (m - 1) ln y = 2 ln x Solve for m: m = 1 + 2 ln(x) / ln(y) Use log rule of product: log4(10) = log4(2) + log4(5) log4(2) = log4(41/2) = 1/2 Use change of base formula to write: log4(5) = log8(5) / log8(4) = b / (2/3) , since log8(4) = 2/3 log4(10) = log4(2) + log4(5) = (1 + 3b) / 2 log6(216) + [ log(42) - log(6) ] / log(49) = log6(63) + log(42/6) / log(72) = 3 + log(7) /2 log(7) = 3 + 1/2 = 7/2 ((3-1 - 9-1) / 6)1/3 = ((1/3 - 1/9) / 6)1/3 = ((6 / 27) / 6)1/3 = 1/3 Use change of base formula: (logxa)(logab) = logxa (logxb / logxa) = logxb 2 = logae a2 = e ln(a2) = ln e 2 ln a = 1 a = e1/2 Use change of base formula using ln to rewrite the given equation as follows ln (x) / ln(3) = A ln(x) / ln(5) A = ln(5) / ln(3) Rewrite given equation as: log [ log (2 + log2(x + 1)) ] = log (1) , since log(1) = 0. log (2 + log2(x + 1)) = 1 2 + log2(x + 1) = 10 log2(x + 1) = 8 x + 1 = 28 x = 28 - 1 Note that b4 logbx = x4 The given equation may be written as: 2x x4 = 486 x = 2431/5 = 3 Group terms and use power rule: ln (x - 1)(2x - 1) = ln (x + 1)2 ln function is a one to one function, hence: (x - 1)(2x - 1) = (x + 1)2 Solve the above quadratic function: x = 0 and x = 5 Only x = 5 is a valid solution to the equation given above since x = 0 is not in the domain of the expressions making the equations. Solve: 0 = 2 log( sqrt(x - 1) - 2) Divide both sides by 2: log( sqrt(x - 1) - 2) = 0 Use the fact that log(1)= 0: sqrt(x - 1) - 2 = 1 Rewrite as: sqrt(x - 1) = 3 Raise both sides to the power 2: (x - 1) = 32 x - 1 = 9 x = 10 Given: 9x - 3x - 8 = 0 Note that: 9x = (3x)2 Equation may be written as: (3x)2 - 3x - 8 = 0 Let y = 3x and rewite equation with y: y2 - y - 8 = 0 Solve for y: y = ( 1 + sqrt(33) ) / 2 and ( 1 - sqrt(33) ) / 2 Since y = 3x, the only acceptable solution is y = ( 1 + sqrt(33) ) / 2 3x = ( 1 + sqrt(33) ) / 2 Use ln on both sides: ln 3x = ln [ ( 1 + sqrt(33) ) / 2] Simplify and solve: x = ln [ ( 1 + sqrt(33) ) / 2] / ln 3 Given: 4x - 2 = 3x + 4 Take ln of both sides: ln ( 4x - 2 ) = ln ( 3x + 4 ) Simplify: (x - 2) ln 4 = (x + 4) ln 3 Expand: x ln 4 - 2 ln 4 = x ln 3 + 4 ln 3 Group like terms: x ln 4 - x ln 3 = 4 ln 3 + 2 ln 4 Solve for x: x = ( 4 ln 3 + 2 ln 4 ) / (ln 4 - ln 3) = ln (34 * 42) / ln (4/3) = ln (34 * 24) / ln (4/3) = 4 ln(6) / ln(4/3) Rewrite the given equation using exponential form: x- 3 / 4 = 1 / 8 Raise both sides of the above equation to the power -4 / 3: (x- 3 / 4)- 4 / 3 = (1 / 8)- 4 / 3 simplify: x = 84 / 3 = 24 = 16

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