Exponential Logarithmic Transformations

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Answer Key & Explanations

Exponential Logarithmic Transformations · Grade 12 · Worksheet 2

1. A solid is formed by rotating the region bounded by the curve y = e^x, the x-axis, the y-axis, and the vertical line x = 2 about the x-axis. Using calculus, determine the volume of this three-dimensional solid. (Express your answer in terms of π and e.) Answer: (π/2)(e^4 - 1) Solution: We are finding the volume of the solid formed by rotating the region bounded by y = e^x, the x-axis, the y-axis, and the vertical line x = 2 about the x-axis.
   Full step-by-step solution

   We are finding the volume of the solid formed by rotating the region bounded by y = e^x, the x-axis, the y-axis, and the vertical line x = 2 about the x-axis. --- **Step 1: Identify the method** When rotating a curve y = f(x) about the x-axis, the volume from x = a to x = b is given by the disk method: Volume = π ∫ [y]^2 dx from a to b. Here, y = e^x, so [y]^2 = (e^x)^2 = e^(2x). The region is bounded on the left by the y-axis (x = 0) and on the right by x = 2. So limits: a = 0, b = 2. --- **Step 2: Set up the integral** Volume = π ∫ from 0 to 2 of [e^(2x)] dx. --- **Step 3: Integrate** We know ∫ e^(2x) dx = (1/2) e^(2x) + C. So: π ∫ from 0 to 2 e^(2x) dx = π [ (1/2) e^(2x) ] from 0 to 2. --- **Step 4: Evaluate the definite integral** First, at x = 2: (1/2) e^(2*2) = (1/2) e^4. At x = 0: (1/2) e^(0) = (1/2) * 1 = 1/2. Subtract: (1/2) e^4 - (1/2) = (1/2)(e^4 - 1). --- **Step 5: Multiply by π** Volume = π * (1/2)(e^4 - 1) = (π/2)(e^4 - 1). --- **Final Answer:** (π/2)(e^4 - 1)

2. Describe the transformations that map f(x) = log₁₀(x) to g(x) = -2 log₁₀(1/6 x + 1) + 1. Answer: Vertical stretch by factor 2, reflection across x-axis, horizontal stretch by factor 6, horizontal shift left by 6 units, vertical shift up by 1 unit Solution: Start with f(x) = log₁₀(x). The coefficient -2 outside the log indicates a vertical stretch by factor 2 and a reflection across the x-axis (since it is negative). Inside the log, we have (1/6 x + 1).
   Full step-by-step solution

   Step 1: Start with f(x) = log₁₀(x). Step 2: The coefficient -2 outside the log indicates a vertical stretch by factor 2 and a reflection across the x-axis (since it is negative). Step 3: Inside the log, we have (1/6 x + 1). Factor out 1/6: 1/6 x + 1 = 1/6 (x + 6). Step 4: The factor 1/6 inside the log means a horizontal stretch by factor 6 (reciprocal of 1/6). Step 5: The +6 inside the parentheses means a horizontal shift left by 6 units (opposite sign). Step 6: The +1 outside the log means a vertical shift up by 1 unit. Step 7: Combining all: vertical stretch by factor 2, reflection across x-axis, horizontal stretch by factor 6, horizontal shift left by 6 units, vertical shift up by 1 unit.

3. A pharmaceutical company is modeling the concentration of a new medication in a patient's bloodstream over time. The concentration C(t) in milligrams per liter is given by the function C(t) = 80e^(-0.15t) - 20e^(-0.4t), where t is time in hours. Determine the time at which the medication concentration reaches its maximum value, and find that maximum concentration. Answer: t = ln(8)/0.25 ≈ 8.32 hours, maximum concentration ≈ 37.89 mg/L Solution: For functions composed of exponential terms, this involves setting the derivative equal to zero and solving the resulting equation.
   Full step-by-step solution

   When modeling real-world phenomena like drug concentration, maximum values occur when the rate of change becomes zero. For functions composed of exponential terms, this involves setting the derivative equal to zero and solving the resulting equation. The solution typically requires logarithmic properties to isolate the time variable, then substituting back to find the maximum value.

4. Describe the transformations that map f(x) = log₆(x) to g(x) = -2·log₆(6x + 36) + 1. Answer: Horizontal shift left 6, horizontal compression by factor 1/6, vertical stretch by factor 2, reflection across x-axis, vertical shift up 1. Solution: Start with g(x) = -2·log₆(6x + 36) + 1. Factor the argument: 6x + 36 = 6(x + 6). So g(x) = -2·log₆(6(x + 6)) + 1.
   Full step-by-step solution

   Step 1: Start with g(x) = -2·log₆(6x + 36) + 1. Step 2: Factor the argument: 6x + 36 = 6(x + 6). So g(x) = -2·log₆(6(x + 6)) + 1. Step 3: Use the logarithm property log₆(6(x+6)) = log₆(6) + log₆(x+6) = 1 + log₆(x+6). Step 4: Substitute: g(x) = -2[1 + log₆(x+6)] + 1 = -2 - 2·log₆(x+6) + 1 = -2·log₆(x+6) - 1. Step 5: Compare to f(x) = log₆(x). - Inside: x replaced by (x+6) → horizontal shift left 6 units. - Coefficient -2 outside: first multiply by 2 (vertical stretch by factor 2), then negative sign reflects across the x-axis. - Then subtract 1: vertical shift down 1 unit. Step 6: The transformations are: horizontal shift left 6, vertical stretch by factor 2, reflection across x-axis, vertical shift down 1. The answer is: Horizontal shift left 6, vertical stretch by factor 2, reflection across x-axis, vertical shift down 1.

5. e^(ln(5) + 2ln(3)) = ? Answer: 45 Solution: Start with e^(ln(5) + 2ln(3)) Use the property ln(a^b) = b*ln(a) to rewrite 2ln(3) as ln(3^2) = ln(9) Now we have e^(ln(5) + ln(9)) Use the property ln(a) + ln(b) = ln(ab) to combine: ln(5) + ln(9) = ln(5×9) = ln(45) Now we have e^(ln(45)) Since e^ln(x) = x, e^(ln(45)) = 45 The answer is 45.
   Full step-by-step solution

   Step 1: Start with e^(ln(5) + 2ln(3)) Step 2: Use the property ln(a^b) = b*ln(a) to rewrite 2ln(3) as ln(3^2) = ln(9) Step 3: Now we have e^(ln(5) + ln(9)) Step 4: Use the property ln(a) + ln(b) = ln(ab) to combine: ln(5) + ln(9) = ln(5×9) = ln(45) Step 5: Now we have e^(ln(45)) Step 6: Since e^ln(x) = x, e^(ln(45)) = 45 The answer is 45.

6. Olivia is analyzing the transformation of an exponential function used to model the growth of a certain investment. The base function is f(x) = 5^x. The transformed function is g(x) = -3 * 5^(x - 10) + 15. Describe all transformations applied to f(x) to obtain g(x), including the direction and magnitude of shifts, any reflections, and any vertical stretches or compressions. Answer: Vertical stretch by a factor of 3, reflection across the x-axis, horizontal shift 10 units to the right, vertical shift 15 units upward. Solution: Write g(x) in transformation form: g(x) = -3 * 5^(x - 10) + 15. This matches the form g(x) = a * b^(x - h) + k, where a = -3, b = 5, h = 10, k = 15. The base function is f(x) = 5^x.
   Full step-by-step solution

   Step 1: Write g(x) in transformation form: g(x) = -3 * 5^(x - 10) + 15. This matches the form g(x) = a * b^(x - h) + k, where a = -3, b = 5, h = 10, k = 15. Step 2: The base function is f(x) = 5^x. The value a = -3 indicates two things: the absolute value |a| = 3 means a vertical stretch by a factor of 3, and the negative sign means a reflection across the x-axis. Step 3: The value h = 10 means a horizontal shift of 10 units to the right (since x - h shifts right). Step 4: The value k = 15 means a vertical shift of 15 units upward. Step 5: Therefore, the transformations from f(x) = 5^x to g(x) = -3 * 5^(x - 10) + 15 are: vertical stretch by a factor of 3, reflection across the x-axis, horizontal shift 10 units to the right, and vertical shift 15 units upward. The answer is: vertical stretch by a factor of 3, reflection across the x-axis, horizontal shift 10 units to the right, vertical shift 15 units upward.

7. A geometric solid is formed by rotating the region bounded by the curve y = sqrt(x), the x-axis, and the vertical line x = 9 about the x-axis. Using the method of disks, determine the volume of this three-dimensional solid. (Express your answer in terms of π.) Answer: 81π/2 Solution: The region is bounded by y = sqrt(x), y = 0 (x-axis), and x = 9. When rotated about the x-axis, each vertical slice becomes a disk with radius r(x) = sqrt(x).
   Full step-by-step solution

   Step 1: The region is bounded by y = sqrt(x), y = 0 (x-axis), and x = 9. Step 2: When rotated about the x-axis, each vertical slice becomes a disk with radius r(x) = sqrt(x). Step 3: The area of each disk is A(x) = π[r(x)]^2 = π[sqrt(x)]^2 = πx. Step 4: The volume is the integral of these disk areas from x = 0 to x = 9. Step 5: V = ∫ from 0 to 9 of πx dx Step 6: V = π ∫ from 0 to 9 of x dx Step 7: V = π [x^2/2] from 0 to 9 Step 8: V = π [(9^2/2) - (0^2/2)] Step 9: V = π [(81/2) - 0] Step 10: V = 81π/2 The answer is 81π/2.

8. Describe the transformations that map f(x) = 5^x to g(x) = -5^(x-2) + 10. Answer: Horizontal shift right 2 units, reflection across the x-axis, vertical shift up 10 units Solution: Write g(x) in transformation form: g(x) = -5^(x-2) + 10. Compare to f(x) = 5^x. Identify the horizontal shift: The exponent is (x - 2), so h = 2.
   Full step-by-step solution

   Step 1: Write g(x) in transformation form: g(x) = -5^(x-2) + 10. Compare to f(x) = 5^x. Step 2: Identify the horizontal shift: The exponent is (x - 2), so h = 2. This shifts the graph 2 units to the right. Step 3: Identify the vertical stretch/reflection: The coefficient in front is -1, so a = -1. Since a is negative, there is a reflection across the x-axis. The absolute value |a| = 1 means no vertical stretch or compression. Step 4: Identify the vertical shift: The constant term is +10, so k = 10. This shifts the graph 10 units upward. Step 5: Combine the transformations: Starting from f(x) = 5^x, shift right 2 units, reflect across the x-axis, then shift up 10 units to obtain g(x). The answer is: horizontal shift right 2 units, reflection across the x-axis, vertical shift up 10 units.