Exponential Logarithmic Transformations

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Exponential Logarithmic Transformations · Grade 12 · Worksheet 3

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. Describe the resulting 3D shape and find its volume using integration. Answer: π/2(e^4 - 1) Solution: y = e^x - x-axis (y = 0) - y-axis (x = 0) - vertical line x = 2 So the 2D region is in the first quadrant, under y = e^x, from x = 0 to x = 2. We rotate this region about the x-axis.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understanding the problem** We have the curve: y = e^x Bounded by: - x-axis (y = 0) - y-axis (x = 0) - vertical line x = 2 So the 2D region is in the first quadrant, under y = e^x, from x = 0 to x = 2. We rotate this region **about the x-axis**. --- **Step 2: Shape formed** When we rotate the curve y = e^x about the x-axis, each vertical segment of length e^x (from y = 0 to y = e^x) sweeps out a **disk** (circle) of radius e^x. So the 3D shape is a **solid of revolution** that looks like a "trumpet" shape, wider as x increases because e^x increases. --- **Step 3: Volume by disks method** For rotation about the x-axis, the cross-section at x is a disk of radius R(x) = e^x. Area of disk = π [R(x)]^2 = π (e^x)^2 = π e^(2x). Volume = integral from x = 0 to x = 2 of [Area] dx. So: V = ∫ from 0 to 2 of π e^(2x) dx. --- **Step 4: Factor out π** V = π ∫ from 0 to 2 e^(2x) dx. --- **Step 5: Integrate e^(2x)** Recall: ∫ e^(ax) dx = (1/a) e^(ax) + C. Here a = 2, so: ∫ e^(2x) dx = (1/2) e^(2x). --- **Step 6: Evaluate definite integral** ∫ from 0 to 2 e^(2x) dx = [ (1/2) e^(2x) ] from 0 to 2 = (1/2) e^(4) - (1/2) e^(0) = (1/2) e^4 - (1/2)(1) = (1/2)(e^4 - 1). --- **Step 7: Multiply by π** V = π × (1/2)(e^4 - 1) = (π/2)(e^4 - 1). --- **Final Answer:** Volume = π/2 (e^4 - 1) --- **Summary:** The solid is a trumpet-like shape from rotating y = e^x about x-axis from x=0 to x=2. We used disk method: radius = e^x, area = π e^(2x), integrated to get π/2 (e^4 - 1).

2. A geometric solid is formed by rotating the region bounded by the curve y = ln(x), the x-axis, and the vertical lines x = 1 and x = 4 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: π(8ln(4)-15/4) Solution: Set up the disk method volume formula: V = π∫[from a to b] [f(x)]² dx For our region: f(x) = ln(x), a = 1, b = 4 V = π∫[from 1 to 4] [ln(x)]² dx Use integration by parts: Let u = [ln(x)]², dv = dx, then du = 2ln(x)/x dx, v = x ∫[ln(x)]² dx = x[ln(x)]² - ∫x*(2ln(x)/x) dx = x[ln(x)]² - 2∫ln(x) dx…
   Full step-by-step solution

   Step 1: Set up the disk method volume formula: V = π∫[from a to b] [f(x)]² dx Step 2: For our region: f(x) = ln(x), a = 1, b = 4 Step 3: V = π∫[from 1 to 4] [ln(x)]² dx Step 4: Use integration by parts: Let u = [ln(x)]², dv = dx, then du = 2ln(x)/x dx, v = x Step 5: ∫[ln(x)]² dx = x[ln(x)]² - ∫x*(2ln(x)/x) dx = x[ln(x)]² - 2∫ln(x) dx Step 6: Evaluate ∫ln(x) dx using integration by parts: Let u = ln(x), dv = dx, then du = 1/x dx, v = x Step 7: ∫ln(x) dx = xln(x) - ∫x*(1/x) dx = xln(x) - ∫dx = xln(x) - x Step 8: Substitute back: ∫[ln(x)]² dx = x[ln(x)]² - 2[xln(x) - x] = x[ln(x)]² - 2xln(x) + 2x Step 9: Evaluate from 1 to 4: [4(ln4)² - 8ln4 + 8] - [1(ln1)² - 2ln1 + 2] Step 10: Since ln1 = 0, this simplifies to: 4(ln4)² - 8ln4 + 8 - 2 = 4(ln4)² - 8ln4 + 6 Step 11: Multiply by π: V = π[4(ln4)² - 8ln4 + 6] Step 12: Note that (ln4)² = (2ln2)² = 4(ln2)², but we can also simplify using integration by parts differently Step 13: Alternative approach: ∫(lnx)²dx = x(lnx)² - 2xlnx + 2x + C Step 14: Evaluate from 1 to 4: [4(ln4)² - 8ln4 + 8] - [0 - 0 + 2] = 4(ln4)² - 8ln4 + 6 Step 15: Since ln4 = 2ln2, we have: 4(2ln2)² - 8(2ln2) + 6 = 16(ln2)² - 16ln2 + 6 Step 16: The answer is π[4(ln4)² - 8ln4 + 6] or equivalently π(16(ln2)² - 16ln2 + 6)

3. log₂(8) + ln(e³) = ? Answer: 6 Solution: log₂(8) + ln(e³) Evaluate log₂(8) log₂(8) means: "2 raised to what power equals 8?" Since 2³ = 8, log₂(8) = 3. ln means natural logarithm, which is log base e.
   Full step-by-step solution

   Let's solve step by step. Step 1: Understand the problem We have: log₂(8) + ln(e³) Step 2: Evaluate log₂(8) log₂(8) means: "2 raised to what power equals 8?" Since 2³ = 8, log₂(8) = 3. Step 3: Evaluate ln(e³) ln means natural logarithm, which is log base e. ln(e³) means: "e raised to what power equals e³?" The answer is 3, because e³ = e³. So ln(e³) = 3. Step 4: Add the results log₂(8) + ln(e³) = 3 + 3 = 6. Step 5: Final answer The correct answer is 6.

4. e^(2ln(3) - ln(2)) = ? Answer: 4.5 Solution: Step 1: Apply the property a·ln(b) = ln(b^a) to 2ln(3): 2ln(3) = ln(3^2) = ln(9) Step 2: Rewrite the expression: e^(ln(9) - ln(2)) Step 3: Apply the property ln(a) - ln(b) = ln(a/b): e^(ln(9/2)) Step 4: Apply the property e^(ln(x)) = x: 9/2 Step 5: Convert to decimal: 9/2 = 4.5 The answer is 4.5.
   Full step-by-step solution

   Step 1: Apply the property a·ln(b) = ln(b^a) to 2ln(3): 2ln(3) = ln(3^2) = ln(9) Step 2: Rewrite the expression: e^(ln(9) - ln(2)) Step 3: Apply the property ln(a) - ln(b) = ln(a/b): e^(ln(9/2)) Step 4: Apply the property e^(ln(x)) = x: 9/2 Step 5: Convert to decimal: 9/2 = 4.5 The answer is 4.5.

5. Dr. Chen is studying bacterial growth in a lab experiment. The population P(t) of bacteria after t hours is modeled by the function P(t) = 500e^(0.15t). Due to limited resources, the growth rate decreases over time, and the actual population follows the modified function P(t) = 500e^(0.15t - 0.002t²). After how many hours will the bacterial population reach its maximum? Answer: 37.5 Solution: P(t) = 500 * e^(0.15t - 0.002t²) To find when the population reaches its maximum, we need to find the value of t that maximizes P(t).
   Full step-by-step solution

   We are given the modified population function: P(t) = 500 * e^(0.15t - 0.002t²) To find when the population reaches its maximum, we need to find the value of t that maximizes P(t). --- **Step 1: Understand the approach** Since P(t) = 500 * e^(exponent) and 500 is a positive constant, maximizing P(t) is equivalent to maximizing the exponent inside: f(t) = 0.15t - 0.002t² This is because e^(f(t)) increases if f(t) increases, and decreases if f(t) decreases. So the maximum of P(t) occurs at the same t as the maximum of f(t). --- **Step 2: Find the critical point of f(t)** We take the derivative of f(t) with respect to t: f'(t) = 0.15 - 2 * 0.002 * t f'(t) = 0.15 - 0.004t Set f'(t) = 0 to find the critical point: 0.15 - 0.004t = 0 0.15 = 0.004t t = 0.15 / 0.004 --- **Step 3: Calculate t** 0.15 / 0.004 = 150 / 4 = 37.5 So t = 37.5 hours. --- **Step 4: Verify it's a maximum** The second derivative of f(t) is: f''(t) = -0.004 Since f''(t) is negative, the critical point is a maximum. --- **Final Answer:** The bacterial population reaches its maximum after 37.5 hours.

6. A pharmaceutical company is modeling the concentration of a new drug 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. The drug is considered effective when the concentration is at least 25 mg/L and safe when it is below 40 mg/L. During what time interval is the drug both effective and safe? Answer: 1.5 < t < 4.2 Solution: When analyzing drug concentration models, we often need to determine time intervals where the concentration stays within therapeutic ranges.
   Full step-by-step solution

   When analyzing drug concentration models, we often need to determine time intervals where the concentration stays within therapeutic ranges. This involves finding where an exponential function falls between two values, which typically requires solving equations using logarithmic properties. The intersection points define the boundaries of the safe and effective window.

7. ∫(2x·e^(x²) + 3x²) dx from 0 to 1 = ? Answer: e + 1 Solution: Split the integral: ∫(2x·e^(x²) + 3x²) dx = ∫2x·e^(x²) dx + ∫3x² dx For ∫2x·e^(x²) dx, use substitution u = x², du = 2x dx This becomes ∫e^u du = e^u = e^(x²) For ∫3x² dx, use power rule: 3·(x³/3) = x³ Evaluate from 0 to 1: [e^(1²) + 1³] - [e^(0²) + 0³] = (e + 1) - (1 + 0) = e + 1 - 1 = e
   Full step-by-step solution

   Step 1: Split the integral: ∫(2x·e^(x²) + 3x²) dx = ∫2x·e^(x²) dx + ∫3x² dx Step 2: For ∫2x·e^(x²) dx, use substitution u = x², du = 2x dx Step 3: This becomes ∫e^u du = e^u = e^(x²) Step 4: For ∫3x² dx, use power rule: 3·(x³/3) = x³ Step 5: So the antiderivative is e^(x²) + x³ Step 6: Evaluate from 0 to 1: [e^(1²) + 1³] - [e^(0²) + 0³] = (e + 1) - (1 + 0) = e + 1 - 1 = e Step 7: The answer is e

8. e^(3ln(2) - ln(4)) = ? Answer: 2 Solution: Start with e^(3ln(2) - ln(4)) Apply the property a*ln(b) = ln(b^a): e^(ln(2^3) - ln(4)) Simplify 2^3: e^(ln(8) - ln(4)) Apply the property ln(a) - ln(b) = ln(a/b): e^(ln(8/4)) Simplify 8/4: e^(ln(2)) Apply e^ln(x) = x: 2 The answer is 2.
   Full step-by-step solution

   Step 1: Start with e^(3ln(2) - ln(4)) Step 2: Apply the property a*ln(b) = ln(b^a): e^(ln(2^3) - ln(4)) Step 3: Simplify 2^3: e^(ln(8) - ln(4)) Step 4: Apply the property ln(a) - ln(b) = ln(a/b): e^(ln(8/4)) Step 5: Simplify 8/4: e^(ln(2)) Step 6: Apply e^ln(x) = x: 2 The answer is 2.