Exponential Logarithmic Transformations

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Answer Key & Explanations

Exponential Logarithmic Transformations · Grade 12 · Worksheet 1

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 the method of disks, determine the volume of this solid of revolution. Answer: π/2(e^4 - 1) Solution: - Curve: y = e^x - x-axis (y = 0) - y-axis (x = 0) - Vertical line x = 2 Rotation is about the x-axis. Volume = π ∫ [R(x)]² dx from x = a to x = b, where R(x) is the radius from the axis of rotation to the curve.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the region and the method** We are rotating the region bounded by: - Curve: y = e^x - x-axis (y = 0) - y-axis (x = 0) - Vertical line x = 2 Rotation is about the **x-axis**. Using the **disk method**: Volume = π ∫ [R(x)]² dx from x = a to x = b, where R(x) is the radius from the axis of rotation to the curve. --- **Step 2: Identify R(x) and limits of integration** Since we rotate around the x-axis, the radius R(x) is the y-value of the curve: R(x) = e^x. Limits along x-axis: from x = 0 to x = 2. --- **Step 3: Set up the integral** Volume V = π ∫ from 0 to 2 of [e^x]² dx = π ∫ from 0 to 2 of e^(2x) dx. --- **Step 4: Integrate** ∫ e^(2x) dx = (1/2) e^(2x) + C. So, V = π [ (1/2) e^(2x) ] from 0 to 2. --- **Step 5: Evaluate the definite integral** At x = 2: (1/2) e^(4) At x = 0: (1/2) e^(0) = 1/2 So: V = π [ (1/2) e^4 - (1/2) ] = π * (1/2) [ e^4 - 1 ] --- **Step 6: Final answer** V = π/2 (e^4 - 1) --- **Final Answer:** π/2(e^4 - 1)

2. A 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. Using the method of disks, determine the volume of this three-dimensional shape. Answer: π/2(e^4 - 1) Solution: We are rotating the region under the curve y = e^x, above the x-axis, from x = 0 to x = 2, around the x-axis.
   Full step-by-step solution

   Step 1: Understand the problem and the disk method. We are rotating the region under the curve y = e^x, above the x-axis, from x = 0 to x = 2, around the x-axis. The method of disks tells us that the volume is the integral of the area of circular disks from the start to the end of the interval. The radius of each disk at a point x is the y-value of the function, which is e^x. Step 2: Write the formula for the volume using the disk method. The area of a circle is A = π * r^2. Here, r = e^x, so the area of a cross-sectional disk is A(x) = π * (e^x)^2. The volume is the integral of these areas from x = 0 to x = 2: Volume = integral from 0 to 2 of [ π * (e^x)^2 ] dx Step 3: Simplify the integrand. (e^x)^2 = e^(2x). So the integral becomes: Volume = π * integral from 0 to 2 of e^(2x) dx Step 4: Find the antiderivative of e^(2x). The derivative of e^(2x) is 2e^(2x). Therefore, the antiderivative of e^(2x) is (1/2)e^(2x). We can check this: derivative of (1/2)e^(2x) = (1/2)*2e^(2x) = e^(2x). Correct. Step 5: Evaluate the definite integral. We compute the antiderivative from 0 to 2: integral from 0 to 2 of e^(2x) dx = [ (1/2)e^(2x) ] evaluated 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 to get the final volume. Volume = π * [ (1/2)(e^4 - 1) ] = (π/2)(e^4 - 1) Final Answer: π/2(e^4 - 1)

3. log₂(16) + ln(e⁴) = ? Answer: 8 Solution: Evaluate log₂(16). Since 2^4 = 16, log₂(16) = 4. Evaluate ln(e⁴).
   Full step-by-step solution

   Step 1: Evaluate log₂(16). Since 2^4 = 16, log₂(16) = 4. Step 2: Evaluate ln(e⁴). Since ln(e^x) = x, ln(e⁴) = 4. Step 3: Add the results: 4 + 4 = 8. The answer is 8.

4. Dr. Rodriguez is studying the decay of a radioactive isotope in a geological sample. The amount A(t) in grams remaining after t years is modeled by the function A(t) = 200e^(-0.00012t). The isotope becomes difficult to detect with standard equipment when the amount drops below 50 grams. After how many years will the isotope become difficult to detect? Round your answer to the nearest year. Answer: 11552 Solution: Set up the equation where A(t) equals the detection threshold: 200e^(-0.00012t) = 50 Divide both sides by 200: e^(-0.00012t) = 50/200 = 0.25 Take the natural logarithm of both sides: ln(e^(-0.00012t)) = ln(0.25) Simplify using logarithm properties: -0.00012t = ln(0.25) Calculate ln(0.25):…
   Full step-by-step solution

   Step 1: Set up the equation where A(t) equals the detection threshold: 200e^(-0.00012t) = 50 Step 2: Divide both sides by 200: e^(-0.00012t) = 50/200 = 0.25 Step 3: Take the natural logarithm of both sides: ln(e^(-0.00012t)) = ln(0.25) Step 4: Simplify using logarithm properties: -0.00012t = ln(0.25) Step 5: Calculate ln(0.25): ln(0.25) ≈ -1.386294361 Step 6: Solve for t: t = -1.386294361 / -0.00012 = 11552.45301 Step 7: Round to the nearest year: t ≈ 11552 years The answer is 11552 years.

5. log₂(3x + 1) - log₂(x - 1) = 2 Answer: 5 Solution: log₂(3x + 1) - log₂(x - 1) = 2 Use the logarithm subtraction rule: log A - log B = log(A/B) log₂( (3x + 1)/(x - 1) ) = 2 Rewrite the logarithmic equation in exponential form.
   Full step-by-step solution

   We start with the equation: log₂(3x + 1) - log₂(x - 1) = 2 Step 1: Use the logarithm subtraction rule: log A - log B = log(A/B) So we have: log₂( (3x + 1)/(x - 1) ) = 2 Step 2: Rewrite the logarithmic equation in exponential form. The equation log₂(something) = 2 means: 2^2 = (3x + 1)/(x - 1) So: 4 = (3x + 1)/(x - 1) Step 3: Multiply both sides by (x - 1) to eliminate the fraction. 4(x - 1) = 3x + 1 Step 4: Expand and solve for x. 4x - 4 = 3x + 1 Subtract 3x from both sides: 4x - 3x - 4 = 1 x - 4 = 1 Add 4 to both sides: x = 5 Step 5: Check the domain of the original logarithmic expressions. Original: log₂(3x + 1) and log₂(x - 1) require: 3x + 1 > 0 → 3x > -1 → x > -1/3 x - 1 > 0 → x > 1 The stricter condition is x > 1. Since x = 5 > 1, it is valid. Also check: 3(5) + 1 = 16 > 0, and 5 - 1 = 4 > 0. Thus the solution is valid. Final answer: x = 5

6. Dr. Rodriguez is studying the decay of a radioactive isotope in a geological sample. The remaining mass M(t) in grams after t years is modeled by the function M(t) = 200e^(-0.00012t). The isotope is considered useful for dating purposes when at least 60% of the original mass remains. Determine how many years will pass before the isotope becomes no longer useful for dating, rounding your answer to the nearest hundred years. Answer: 4300 Solution: Identify the original mass: M(0) = 200 grams Calculate 60% of original mass: 0.60 × 200 = 120 grams Set up the equation: 200e^(-0.00012t) = 120 Divide both sides by 200: e^(-0.00012t) = 0.6 Take natural logarithm of both sides: ln(e^(-0.00012t)) = ln(0.6) Simplify using logarithm properties:…
   Full step-by-step solution

   Step 1: Identify the original mass: M(0) = 200 grams Step 2: Calculate 60% of original mass: 0.60 × 200 = 120 grams Step 3: Set up the equation: 200e^(-0.00012t) = 120 Step 4: Divide both sides by 200: e^(-0.00012t) = 0.6 Step 5: Take natural logarithm of both sides: ln(e^(-0.00012t)) = ln(0.6) Step 6: Simplify using logarithm properties: -0.00012t = ln(0.6) Step 7: Calculate ln(0.6) ≈ -0.510826 Step 8: Solve for t: t = -0.510826 / -0.00012 Step 9: Calculate: t ≈ 4256.88 years Step 10: Round to nearest hundred years: 4300 years The answer is 4300 years.

7. A geometric transformation is applied to the graph of the function f(x) = 2^x. The transformation consists of a vertical stretch by a factor of 3, followed by a horizontal translation 2 units to the left, and finally a reflection across the x-axis. Describe the transformed function g(x) in terms of its equation. Answer: g(x) = -3 * 2^(x+2) Solution: f(x) = 2^x Vertical stretch by a factor of 3 A vertical stretch by a factor of 3 multiplies the entire function by 3.
   Full step-by-step solution

   Let's go step by step. We start with the original function: f(x) = 2^x --- **Step 1: Vertical stretch by a factor of 3** A vertical stretch by a factor of 3 multiplies the entire function by 3. So: h1(x) = 3 * f(x) = 3 * 2^x --- **Step 2: Horizontal translation 2 units to the left** A horizontal shift to the left by 2 units means we replace x with (x + 2) in the function. So: h2(x) = h1(x + 2) = 3 * 2^(x + 2) --- **Step 3: Reflection across the x-axis** Reflection across the x-axis multiplies the entire function by -1. So: g(x) = - h2(x) = - [ 3 * 2^(x + 2) ] --- **Final transformed function:** g(x) = -3 * 2^(x + 2) --- **Answer:** g(x) = -3 * 2^(x+2)

8. The function f(x) = 4^(x-2) is transformed to g(x) = -3f(2x+1) + 5. If the point (3,4) lies on the graph of f(x), what is the corresponding y-coordinate of the transformed point on g(x)? Answer: -7 Solution: The original point on f(x) is (3,4), meaning f(3) = 4 For the transformation g(x) = -3f(2x+1) + 5, we need to find x such that 2x+1 = 3 Solve 2x+1 = 3 2x = 2 x = 1 Now evaluate g(1) = -3f(2(1)+1) + 5 = -3f(3) + 5 Since f(3) = 4, substitute: g(1) = -3(4) + 5 = -12 + 5 = -7 The answer is -7.
   Full step-by-step solution

   Step 1: The original point on f(x) is (3,4), meaning f(3) = 4 Step 2: For the transformation g(x) = -3f(2x+1) + 5, we need to find x such that 2x+1 = 3 Step 3: Solve 2x+1 = 3 2x = 2 x = 1 Step 4: Now evaluate g(1) = -3f(2(1)+1) + 5 = -3f(3) + 5 Step 5: Since f(3) = 4, substitute: g(1) = -3(4) + 5 = -12 + 5 = -7 The answer is -7.