Math Logs Test Review Solution

Functions and Operations: f(g(5)) = 1 g(h(3)) = No Solution h(f(6)) = -10 f(g(x)) = x - 5, D: [5, ∞) f(h(x)) = (6-x)²/9, D: (-∞, ∞) g(h(x)) = √(-3-x/3), D: (-∞, -9) g(f(x)) = √(x-5), D: (-∞, -√5) U (√5, ∞) h(f(x)) = (6-x²)/3, D: (-∞, ∞) h(g(x)) = (6-√(x-5))/3, D: [5, ∞) Inverse Functions: f(x) = x² + 17: No inverse function f(x) = 2x - 18 + π: Yes, inverse function exists f(x) = 5x⁴ + 17x - 8: No inverse function Finding Inverse Functions: f(x) = 2(x+5)^(4/3) Inverse: f⁻¹(x) = √(x/2) - 5 f(x) = (3/5)x + 8 Inverse: f⁻¹(x) = (5/3)(x-8) Function and Graph: Function: f(x) = (2/3)x³ - 5 Table of values: x | y -2 | -10.3 -1 | -5.7 0 | -5 1 | -4.3 2 | 1 3 | 5.7 4 | 15 Graph: A cubic function with a positive slope, passing through the points from the table. Inverse Function: Table of values: (reverse the x and y values from the original table) Graph: The inverse function is the reflection of the original function across the line y = x. Inverse Function (Algebraically): x = (2/3)y³ - 5 x + 5 = (2/3)y³ (3/2)(x + 5) = y³ f⁻¹(x) = ∛((3x + 15)/2) Logarithms: log₉27 = x 9^x = 27 3^(2x) = 3^3 2x = 3 x = 3/2 log₃√(27) = x 3^x = 3^(3/4) x = 3/4 log₈4√(32) = v 8^v = 4 * 32^(1/2) 2^(3v) = 2^2 * 2^(5/2) 3v = 9/2 v = 3/2 log(1/10,000) log₁₀(10⁻⁴) -4 Equations: log₈x = 5/2 8^(5/2) = x 2^(3 * 5/2) = x x = 2^(15/2) log₃x = 1/2 3^(1/2) = x x = √3 log₂x = 27 2^27 = x x = 2^27 logₓ125 = -1/2 x^(-1/2) = 125 x = 1/√125 x = 1/5√5 4 - 2log₆a log₆(4/a²) 2log₃m - (1/2)log₃n - 3 - log₃2 log₃(m²/√n) - log₃(27 * 2) log₃(m²/54√n) Simplifying Logarithmic Expressions: log₅(1/250) + 3 + log₅2 log₅(1/250) + log₅(5³) + log₅2 log₅(5³/250 * 2) log₅(1) 0 (1/6)(2log₉4 + 2log₈2) (1/6)(log₉16 + log₈4) (1/6)(log₉(64)) (1/6)(2) 1/3 Solving Logarithmic Equations: 3 + 2ln(x) = 10 2ln(x) = 7 ln(x) = 7/2 e^(7/2) = x x ≈ 331.15 e^x = 413 ln(e^x) = ln(413) x = ln(413) x ≈ 6.025 log₃(3x) = log₃3 + log₃x log₃(3x) = log₃(3x) 3x = 3x (0, ∞) 3log₃(3x+2) = log₃(6-x) log₃((3x+2)³) = log₃(6-x) (3x+2)³ = 6-x 27x³ + 54x² + 36x + 8 = 6 - x 27x³ + 54x² + 37x + 2 = 0 (x+1)(9x² + 45x + 2) = 0 x = -1 log₄x - log₄(x-1) = 1/2 log₄(x/(x-1)) = 1/2 4^(1/2) = x/(x-1) 2 = x/(x-1) 2x - 2 = x x = 2 log₃3 + log₃x = log₅5 + log₅(x-2) log₃(3x) = log₅(5x-10) 3x = 5x-10 -2x = -10 x = 5 log₆(2x-5) - log₆(7x+10) = -1 log₆((2x-5)/(7x+10)) = -1 6^(-1) = (2x-5)/(7x+10) 1/6 = (2x-5)/(7x+10) 7x+10 = 12x-30 5x = 40 x = 8 log(10x) - log(2+√x) = 1 log(10x/(2+√x)) = 1 10 = 10x/(2+√x) 20+10√x = 10x 10√x = 10x-20 √x = x-2 x = (x-2)² x² - 4x + 4 = x x² - 5x + 4 = 0 (x-4)(x-1) = 0 x = 4 or x = 1 ln(x-1) + ln(x+2) = 1 ln((x-1)(x+2)) = 1 e = (x-1)(x+2) e = x² + x - 2 x² + x - 2 - e = 0 Using the quadratic formula: x = (-1 ± √(1² - 4(1)(-2-e))) / (2*1) x = (-1 ± √(1 + 8 + 4e)) / 2 x = (-1 ± √(9 + 4e)) / 2 x ≈ 1.718 or x ≈ -2.718 Solving Exponential Equations: 4^(x-3) = (1/2)^x 2^(2(x-3)) = 2^(-x) 2x - 6 = -x 3x = 6 x = 2 (5^4)^x = (25^5)(5^(5x-1)) (5^4)^x = (5^10)(5^(5x-1)) 5^(4x) = 5^(10+5x-1) 4x = 9 + 5x -x = 9 x = -9 (81^(3-x))^(1/2) = (1/9)^(5x-6) * √27^x (3^4)^(3-x) = (3^-2)^(5x-6) * 3^(5/2) 3^(12-4x) = 3^(-10x+12) * 3^(5/2) 12-4x = -10x+12 + 5/2 6x = 5/2 x = 5/12 2^(2x) = (2^(-1))^(x-3) 2^(2x) = 2^(-x+3) 2x = -x + 3 3x = 3 x = 1 e^(2-3x) = 12 ln(e^(2-3x)) = ln(12) 2-3x = ln(12) -3x = ln(12) - 2 x = (-ln(12) + 2)/3 x ≈ -0.162 4e^(2x) = 7 e^(2x) = 7/4 ln(e^(2x)) = ln(7/4) 2x = ln(7/4) x = ln(7/4)/2 x ≈ 0.559 50 = 4 + 4e^(-x) 46 = 4e^(-x) 11.5 = e^(-x) ln(11.5) = -x x = -ln(11.5) x ≈ -2.442 (6^(1/2))^(2x+9) = (6^2)^(3x-1) 6^(x+9/2) = 6^(6x-2) x+9/2 = 6x-2 13/2 = 5x x = 13/10 e^(2-3x) = 12 ln(e^(2-3x)) = ln(12) 2-3x = ln(12) -3x = ln(12) - 2 x = (-ln(12) + 2)/3 x ≈ -0.162 e^(2x) = 7/4 ln(e^(2x)) = ln(7/4) 2x = ln(7/4) x = ln(7/4)/2 x ≈ 0.559 2^(1-x) = 3 ln(2^(1-x)) = ln(3) (1-x)ln(2) = ln(3) 1-x = ln(3)/ln(2) x = 1 - ln(3)/ln(2) x ≈ 0.585 12^x = 5^(x+4) log(12^x) = log(5^(x+4)) xlog(12) = (x+4)log(5) xlog(12) - xlog(5) = 4log(5) x(log(12)-log(5)) = 4log(5) x = 4log(5) / (log(12)-log(5)) x ≈ 7.353 4 + 3^(5x) = 8 3^(5x) = 4 ln(3^(5x)) = ln(4) 5xln(3) = ln(4) x = ln(4)/(5ln(3)) x ≈ 0.252 50 = 4 + 4e^(-x) 46 = 4e^(-x) 11.5 = e^(-x) ln(11.5) = -x x = -ln(11.5) x ≈ -2.442 100(1.04)^(2x) = 300 (1.04)^(2x) = 3 ln((1.04)^(2x)) = ln(3) 2xln(1.04) = ln(3) x = ln(3)/(2ln(1.04)) x ≈ 14.006 x^2 * 2^x - 2^x = 0 2^x(x^2 - 1) = 0 2^x = 0 (no solution) or x^2 - 1 = 0 x = ±1 4x^3e^(-3x) - 3x^4e^(-3x) = 0 e^(-3x)(4x^3 - 3x^4) = 0 e^(-3x) = 0 (no solution) or 4x^3 - 3x^4 = 0 x^3(4 - 3x) = 0 x = 0 or 4 - 3x = 0 x = 0 or x = 4/3 e^(2x) - 3e^x + 2 = 0 (e^x - 1)(e^x - 2) = 0 e^x = 1 or e^x = 2 x = ln(1) or x = ln(2) x = 0 or x ≈ 0.693 e^(4x) + 4e^(2x) - 21 = 0 (e^(2x) + 7)(e^(2x) - 3) = 0 e^(2x) = -7 (no solution) or e^(2x) = 3 2ln(e) = ln(3) 2x = ln(3) x = ln(3)/2 x ≈ 0.549