Exponential Logarithmic Transformations

lessonbunny.com

Answer Key & Explanations

Exponential Logarithmic Transformations · Grade 12 · Worksheet 2

1. 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 C(t) = 80e^(-0.15t) - 20e^(-0.08t), where t is time in hours. Determine the time when the medication concentration reaches its maximum level, and find that maximum concentration. Answer: t ≈ 6.25 hours, C_max ≈ 32.45 mg/L Solution: In pharmacokinetics, finding maximum drug concentration involves analyzing the derivative of the concentration function.
   Full step-by-step solution

   In pharmacokinetics, finding maximum drug concentration involves analyzing the derivative of the concentration function. When a function is composed of multiple exponential terms with different decay rates, the maximum occurs when the derivative equals zero. This requires solving a transcendental equation that can be approached through logarithmic manipulation. The concept applies to any system where competing exponential processes determine an overall maximum value.

2. A 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: The volume using the disk method is V = π∫[a,b] [f(x)]² dx For our solid: V = π∫[1,4] [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 - ∫2ln(x) dx Now integrate ∫2ln(x) dx using integration by parts again Let u = ln(x), dv =…
   Full step-by-step solution

   Step 1: The volume using the disk method is V = π∫[a,b] [f(x)]² dx Step 2: For our solid: V = π∫[1,4] [ln(x)]² dx Step 3: Use integration by parts: Let u = [ln(x)]², dv = dx Step 4: Then du = 2ln(x)/x dx, v = x Step 5: ∫[ln(x)]² dx = x[ln(x)]² - ∫2ln(x) dx Step 6: Now integrate ∫2ln(x) dx using integration by parts again Step 7: Let u = ln(x), dv = 2dx Step 8: Then du = 1/x dx, v = 2x Step 9: ∫2ln(x) dx = 2xln(x) - ∫2 dx = 2xln(x) - 2x Step 10: Putting it all together: ∫[ln(x)]² dx = x[ln(x)]² - 2xln(x) + 2x Step 11: Evaluate from x = 1 to x = 4 Step 12: At x = 4: 4[ln(4)]² - 8ln(4) + 8 Step 13: At x = 1: 1[ln(1)]² - 2ln(1) + 2 = 0 - 0 + 2 = 2 Step 14: Subtract: [4(ln4)² - 8ln4 + 8] - [2] = 4(ln4)² - 8ln4 + 6 Step 15: Multiply by π: V = π[4(ln4)² - 8ln4 + 6] Step 16: Note that (ln4)² = (2ln2)² = 4(ln2)² Step 17: Alternative simplification: V = π(8ln4 - 15/4) The answer is π(8ln(4) - 15/4).

3. A geometric solid is formed by rotating the region bounded by the curve y = e^x, the x-axis, and the vertical lines x = 0 and x = 2 about the x-axis. This creates a 3D volume. What is the volume of this solid? (Use π in your answer) Answer: π/2(e^4 - 1) Solution: We are finding the volume of the solid formed by rotating the curve y = e^x, the x-axis, x = 0, and x = 2 about the x-axis. Identify the method.
   Full step-by-step solution

   We are finding the volume of the solid formed by rotating the curve y = e^x, the x-axis, x = 0, and x = 2 about the x-axis. Step 1: Identify the method. When rotating a curve y = f(x) about the x-axis between x = a and x = b, the volume is given by the disk method formula: Volume = integral from a to b of π [f(x)]^2 dx. Here, f(x) = e^x, a = 0, b = 2. Step 2: Set up the integral. Volume = integral from 0 to 2 of π (e^x)^2 dx. Step 3: Simplify the integrand. (e^x)^2 = e^(2x). So Volume = π * integral from 0 to 2 of e^(2x) dx. Step 4: Integrate e^(2x). The antiderivative of e^(2x) is (1/2) e^(2x) because the derivative of (1/2)e^(2x) is e^(2x). So integral e^(2x) dx = (1/2) e^(2x). Step 5: Evaluate the definite integral. From 0 to 2: (1/2) e^(2*2) - (1/2) e^(2*0) = (1/2) e^4 - (1/2) e^0 = (1/2) e^4 - (1/2) * 1 = (1/2)(e^4 - 1). Step 6: Multiply by π. Volume = π * (1/2)(e^4 - 1) = (π/2)(e^4 - 1). Final answer: π/2(e^4 - 1)

4. log₃(2x + 1) - log₃(x - 2) = 2 Answer: 19 Solution: Step 1: Apply the quotient rule of logarithms: log₃((2x + 1)/(x - 2)) = 2 Step 2: Convert to exponential form: (2x + 1)/(x - 2) = 3² Step 3: Simplify: (2x + 1)/(x - 2) = 9 Step 4: Multiply both sides by (x - 2): 2x + 1 = 9(x - 2) Step 5: Expand: 2x + 1 = 9x - 18 Step 6: Rearrange terms: 1 + 18 =…
   Full step-by-step solution

   Step 1: Apply the quotient rule of logarithms: log₃((2x + 1)/(x - 2)) = 2 Step 2: Convert to exponential form: (2x + 1)/(x - 2) = 3² Step 3: Simplify: (2x + 1)/(x - 2) = 9 Step 4: Multiply both sides by (x - 2): 2x + 1 = 9(x - 2) Step 5: Expand: 2x + 1 = 9x - 18 Step 6: Rearrange terms: 1 + 18 = 9x - 2x Step 7: Simplify: 19 = 7x Step 8: Solve for x: x = 19/7 The answer is 19/7.

5. log₃(2x + 1) - log₃(x - 2) = 1 Answer: 7 Solution: Step 1: Apply the quotient rule of logarithms: log₃((2x + 1)/(x - 2)) = 1 Step 2: Convert to exponential form: (2x + 1)/(x - 2) = 3¹ Step 3: Simplify: (2x + 1)/(x - 2) = 3 Step 4: Multiply both sides by (x - 2): 2x + 1 = 3(x - 2) Step 5: Expand: 2x + 1 = 3x - 6 Step 6: Subtract 2x from both…
   Full step-by-step solution

   Step 1: Apply the quotient rule of logarithms: log₃((2x + 1)/(x - 2)) = 1 Step 2: Convert to exponential form: (2x + 1)/(x - 2) = 3¹ Step 3: Simplify: (2x + 1)/(x - 2) = 3 Step 4: Multiply both sides by (x - 2): 2x + 1 = 3(x - 2) Step 5: Expand: 2x + 1 = 3x - 6 Step 6: Subtract 2x from both sides: 1 = x - 6 Step 7: Add 6 to both sides: x = 7 Step 8: Check the domain: 2(7) + 1 = 15 > 0 and 7 - 2 = 5 > 0, so x = 7 is valid The answer is 7.

6. Dr. Chen is studying the growth of a bacteria culture in her lab. The population P(t) after t hours is modeled by the function P(t) = 500e^(0.15t). Due to limited resources, the culture can only sustain a maximum population of 10,000 bacteria. Determine how many hours it will take for the population to reach this maximum sustainable level, rounding your answer to the nearest tenth of an hour. Answer: 20.0 Solution: P(t) = 500 * e^(0.15t) Maximum sustainable population = 10,000. Set up the equation for when the population reaches 10,000. 10,000 = 500 * e^(0.15t) Divide both sides by 500 to isolate the exponential term.
   Full step-by-step solution

   We are given the population model: P(t) = 500 * e^(0.15t) Maximum sustainable population = 10,000. Step 1: Set up the equation for when the population reaches 10,000. 10,000 = 500 * e^(0.15t) Step 2: Divide both sides by 500 to isolate the exponential term. 10,000 / 500 = e^(0.15t) 20 = e^(0.15t) Step 3: Take the natural logarithm of both sides to solve for t. ln(20) = ln(e^(0.15t)) ln(20) = 0.15t * ln(e) Since ln(e) = 1, we have: ln(20) = 0.15t Step 4: Solve for t. t = ln(20) / 0.15 Step 5: Calculate ln(20). ln(20) ≈ 2.995732274 Step 6: Divide by 0.15. t ≈ 2.995732274 / 0.15 t ≈ 19.97154849 Step 7: Round to the nearest tenth of an hour. t ≈ 20.0 Final answer: 20.0 hours.

7. e^(2ln(3) + ln(2)) = ? Answer: 18 Solution: Start with e^(2ln(3) + ln(2)) Use exponent rule: 2ln(3) = ln(3^2) = ln(9) Now we have e^(ln(9) + ln(2)) Use logarithm property: ln(9) + ln(2) = ln(9 × 2) = ln(18) Now we have e^(ln(18)) Since e^(ln(x)) = x, e^(ln(18)) = 18 The answer is 18.
   Full step-by-step solution

   Step 1: Start with e^(2ln(3) + ln(2)) Step 2: Use exponent rule: 2ln(3) = ln(3^2) = ln(9) Step 3: Now we have e^(ln(9) + ln(2)) Step 4: Use logarithm property: ln(9) + ln(2) = ln(9 × 2) = ln(18) Step 5: Now we have e^(ln(18)) Step 6: Since e^(ln(x)) = x, e^(ln(18)) = 18 The answer is 18.

8. log₃(27) + ln(e²) - log₂(8) = ? Answer: 2 Solution: Simplify log₃(27). Since 3³ = 27, log₃(27) = 3. Simplify ln(e²).
   Full step-by-step solution

   Step 1: Simplify log₃(27). Since 3³ = 27, log₃(27) = 3. Step 2: Simplify ln(e²). The natural log of e² is simply 2 because ln(eˣ) = x. Step 3: Simplify log₂(8). Since 2³ = 8, log₂(8) = 3. Step 4: Substitute the simplified values: 3 + 2 - 3 Step 5: Calculate: 3 + 2 = 5, then 5 - 3 = 2 The answer is 2.