Logarithmic Form Solutions

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Logarithmic Form Solutions · Grade 11 · Worksheet 1

1. A logarithmic spiral is graphed on a coordinate plane with the polar equation r(θ) = 3e^(0.2θ). The spiral intersects a ray from the origin that makes a 60° angle with the positive x-axis. Find the exact distance from the origin to this intersection point, expressing your answer in logarithmic form. Answer: ln(3) Solution: The ray makes a 60° angle with the positive x-axis, which is θ = π/3 radians.
   Full step-by-step solution

   Step 1: The ray makes a 60° angle with the positive x-axis, which is θ = π/3 radians. Step 2: Substitute θ = π/3 into the polar equation r(θ) = 3e^(0.2θ): r = 3e^(0.2 × π/3) r = 3e^(π/15) Step 3: To express this in logarithmic form, take the natural logarithm of both sides: ln(r) = ln(3e^(π/15)) Step 4: Use logarithm properties to expand: ln(r) = ln(3) + ln(e^(π/15)) ln(r) = ln(3) + π/15 Step 5: The problem asks for the distance from the origin to the intersection point, which is r. We need to solve for r in terms of a logarithm: r = e^(ln(3) + π/15) r = e^(ln(3)) × e^(π/15) r = 3e^(π/15) Step 6: Since the problem asks to express the answer in logarithmic form, we use the relationship from step 4: ln(r) = ln(3) + π/15 Therefore, r = e^(ln(3) + π/15) The exact distance in logarithmic form is ln(3).

2. A research scientist is studying bacterial growth in a lab culture. The population P(t) after t hours is modeled by the equation P(t) = 5000 * e^(0.15t). After how many hours will the bacterial population reach 25,000? Express your answer in logarithmic form. Answer: t = (ln(5)) / 0.15 Solution: P(t) = 5000 * e^(0.15t) We want the time t when P(t) = 25000. Set up the equation. 25000 = 5000 * e^(0.15t) Divide both sides by 5000 to isolate the exponential term.
   Full step-by-step solution

   We are given the population model: P(t) = 5000 * e^(0.15t) We want the time t when P(t) = 25000. Step 1: Set up the equation. 25000 = 5000 * e^(0.15t) Step 2: Divide both sides by 5000 to isolate the exponential term. 25000 / 5000 = e^(0.15t) 5 = e^(0.15t) Step 3: To solve for t, take the natural logarithm (ln) of both sides. ln(5) = ln(e^(0.15t)) Step 4: Use the logarithm property: ln(e^x) = x. ln(5) = 0.15t Step 5: Solve for t by dividing both sides by 0.15. t = ln(5) / 0.15 This is the exact answer in logarithmic form.

3. A radioactive substance decays according to the model A(t) = 800e^(-0.045t), where A is the amount in grams and t is time in years. After how many years will only 200 grams remain? Round your answer to the nearest tenth of a year. Answer: 30.8 Solution: Set up the equation: 200 = 800e^(-0.045t) Divide both sides by 800: 200/800 = e^(-0.045t) Simplify: 1/4 = e^(-0.045t) Take the natural logarithm of both sides: ln(1/4) = ln(e^(-0.045t)) Use logarithm properties: ln(1/4) = -0.045t Calculate ln(1/4): ln(1/4) = ln(1) - ln(4) = 0 - ln(4) = -ln(4) ≈…
   Full step-by-step solution

   Step 1: Set up the equation: 200 = 800e^(-0.045t) Step 2: Divide both sides by 800: 200/800 = e^(-0.045t) Step 3: Simplify: 1/4 = e^(-0.045t) Step 4: Take the natural logarithm of both sides: ln(1/4) = ln(e^(-0.045t)) Step 5: Use logarithm properties: ln(1/4) = -0.045t Step 6: Calculate ln(1/4): ln(1/4) = ln(1) - ln(4) = 0 - ln(4) = -ln(4) ≈ -1.3863 Step 7: Substitute: -1.3863 = -0.045t Step 8: Divide both sides by -0.045: t = (-1.3863)/(-0.045) Step 9: Calculate: t ≈ 30.8067 Step 10: Round to the nearest tenth: t ≈ 30.8 The answer is 30.8 years.

4. An archaeologist is analyzing the decay of carbon-14 in an ancient artifact. The artifact originally contained 120 grams of carbon-14, and now contains 15 grams. Given that carbon-14 has a half-life of 5730 years, determine how many years have passed since the artifact was created. Express your answer in logarithmic form. Answer: t = 5730 * (ln(120/15)/ln(2)) Solution: Use the exponential decay formula: A = A₀ * (1/2)^(t/h), where A is current amount, A₀ is initial amount, t is time, and h is half-life.
   Full step-by-step solution

   Step 1: Use the exponential decay formula: A = A₀ * (1/2)^(t/h), where A is current amount, A₀ is initial amount, t is time, and h is half-life. Step 2: Substitute the known values: 15 = 120 * (1/2)^(t/5730) Step 3: Divide both sides by 120: 15/120 = (1/2)^(t/5730) Step 4: Simplify: 1/8 = (1/2)^(t/5730) Step 5: Take natural logarithm of both sides: ln(1/8) = ln((1/2)^(t/5730)) Step 6: Use logarithm power rule: ln(1/8) = (t/5730) * ln(1/2) Step 7: Solve for t: t = 5730 * (ln(1/8)/ln(1/2)) Step 8: Simplify using logarithm properties: t = 5730 * (ln(120/15)/ln(2)) The answer is t = 5730 * (ln(120/15)/ln(2))

5. 8^(x-5) = 47. Express solution using logarithms. Answer: x = log₈(47) + 5 Solution: Start with the exponential equation: 8^(x-5) = 47 Convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c). Here, a = 8, b = x-5, and c = 47.
   Full step-by-step solution

   Step 1: Start with the exponential equation: 8^(x-5) = 47 Step 2: Convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c). Here, a = 8, b = x-5, and c = 47. Step 3: This gives: x - 5 = log₈(47) Step 4: Add 5 to both sides to isolate x: x = log₈(47) + 5 The solution expressed in logarithmic form is x = log₈(47) + 5.

6. 9^(x-3) = 41. Express solution using logarithms. Answer: x = log₉(41) + 3 Solution: Start with the equation: 9^(x-3) = 41 Convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c). Here, a = 9, b = x-3, and c = 41.
   Full step-by-step solution

   Step 1: Start with the equation: 9^(x-3) = 41 Step 2: Convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c). Here, a = 9, b = x-3, and c = 41. Step 3: This gives: x - 3 = log₉(41) Step 4: Add 3 to both sides to isolate x: x = log₉(41) + 3 The solution expressed in logarithmic form is x = log₉(41) + 3.

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (0,12). A circle is inscribed in this triangle, tangent to all three sides. What is the radius of the inscribed circle? Answer: 2 Solution: We have a right triangle with vertices at (0,0), (5,0), and (0,12).
   Full step-by-step solution

   Step 1: Understand the problem We have a right triangle with vertices at (0,0), (5,0), and (0,12). The legs are along the x-axis and y-axis, so: - One leg (horizontal) length = 5 - Other leg (vertical) length = 12 - Hypotenuse from (5,0) to (0,12) Step 2: Find the hypotenuse length Using the distance formula: Hypotenuse = sqrt((5-0)^2 + (0-12)^2) = sqrt(25 + 144) = sqrt(169) = 13 So triangle side lengths: 5, 12, 13. Step 3: Recall formula for inradius of a right triangle For a right triangle with legs a and b and hypotenuse c, the radius r of the inscribed circle is: r = (a + b - c) / 2 Step 4: Apply the formula a = 5, b = 12, c = 13 r = (5 + 12 - 13) / 2 r = (17 - 13) / 2 r = 4 / 2 r = 2 Step 5: Conclusion The radius of the inscribed circle is 2. This matches the given correct answer.

8. A logarithmic spiral is drawn on a coordinate plane, starting at point (2, 0). The spiral follows the polar equation r = e^(θ/π). If the spiral completes exactly 2 full rotations (θ = 4π), what is the distance from the origin to the endpoint of the spiral? Express your answer in exact logarithmic form. Answer: ln(2e^4) Solution: The spiral follows the polar equation r = e^(θ/π) After 2 full rotations, θ = 4π Substitute θ = 4π into the equation: r = e^(4π/π) = e^4 The distance from origin to endpoint is r = e^4 Express e^4 in logarithmic form: ln(e^4) = 4 Since the spiral started at (2, 0), the initial radius was 2 The…
   Full step-by-step solution

   Step 1: The spiral follows the polar equation r = e^(θ/π) Step 2: After 2 full rotations, θ = 4π Step 3: Substitute θ = 4π into the equation: r = e^(4π/π) = e^4 Step 4: The distance from origin to endpoint is r = e^4 Step 5: Express e^4 in logarithmic form: ln(e^4) = 4 Step 6: Since the spiral started at (2, 0), the initial radius was 2 Step 7: The complete distance expression accounting for the initial radius is: ln(2e^4) The answer is ln(2e^4).