Logarithmic Form Solutions

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Logarithmic Form Solutions · Grade 11 · Worksheet 2

1. Matiu is studying the growth of a rare plant species in a controlled environment. The height H(t) of the plant in centimeters after t weeks is modeled by the exponential function H(t) = 12 × 3^(t/4). Matiu needs to determine after how many weeks the plant will reach a height of 108 centimeters. Express your answer in logarithmic form. Answer: t = 4 × log₃(9) or t = 4 × (log(9)/log(3)) Solution: Set up the equation for when the height reaches 108 cm: 12 × 3^(t/4) = 108 Divide both sides by 12: 3^(t/4) = 9 Take the logarithm of both sides (any base): log(3^(t/4)) = log(9) Apply the power rule of logarithms: (t/4) × log(3) = log(9) Multiply both sides by 4: t × log(3) = 4 × log(9) Divide…
   Full step-by-step solution

   Step 1: Set up the equation for when the height reaches 108 cm: 12 × 3^(t/4) = 108 Step 2: Divide both sides by 12: 3^(t/4) = 9 Step 3: Take the logarithm of both sides (any base): log(3^(t/4)) = log(9) Step 4: Apply the power rule of logarithms: (t/4) × log(3) = log(9) Step 5: Multiply both sides by 4: t × log(3) = 4 × log(9) Step 6: Divide both sides by log(3): t = 4 × log(9)/log(3) Step 7: Since 9 = 3^2, we can simplify: t = 4 × log₃(9) = 4 × 2 = 8 The answer in logarithmic form is t = 4 × (log(9)/log(3)) or t = 4 × log₃(9).

2. 7^(2x) = 32. Express solution using logarithms Answer: x = log₇(32)/2 Solution: Start with the equation 7^(2x) = 32 Take the logarithm of both sides with base 7: 2x = log₇(32) Solve for x by dividing both sides by 2: x = log₇(32)/2 The solution in logarithmic form is x = log₇(32)/2
   Full step-by-step solution

   Step 1: Start with the equation 7^(2x) = 32 Step 2: Take the logarithm of both sides with base 7: 2x = log₇(32) Step 3: Solve for x by dividing both sides by 2: x = log₇(32)/2 The solution in logarithmic form is x = log₇(32)/2

3. A logarithmic spiral is drawn on a coordinate plane starting at point (2, 0). The spiral follows 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: 3e^(π/15) Solution: Convert the angle from degrees to radians: 60° = π/3 radians The ray from the origin at angle π/3 has equation θ = π/3 in polar coordinates Substitute θ = π/3 into the spiral equation: r(π/3) = 3e^(0.2 × π/3) Simplify the exponent: 0.2 × π/3 = (1/5) × π/3 = π/15 The distance is r = 3e^(π/15) The…
   Full step-by-step solution

   Step 1: Convert the angle from degrees to radians: 60° = π/3 radians Step 2: The ray from the origin at angle π/3 has equation θ = π/3 in polar coordinates Step 3: Substitute θ = π/3 into the spiral equation: r(π/3) = 3e^(0.2 × π/3) Step 4: Simplify the exponent: 0.2 × π/3 = (1/5) × π/3 = π/15 Step 5: The distance is r = 3e^(π/15) The exact distance from the origin to the intersection point is 3e^(π/15).

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A circle is inscribed within this triangle, tangent to all three sides. What is the radius of the inscribed circle? Answer: 1 Solution: A = (0,0) B = (4,0) C = (4,3) - AB along the x-axis from (0,0) to (4,0), length = 4 - BC vertical from (4,0) to (4,3), length = 3 - AC is the hypotenuse from (0,0) to (4,3), length = sqrt( (4-0)^2 + (3-0)^2 ) = sqrt(16 + 9) = sqrt(25) = 5 a = BC = 3 (opposite angle A) b = AC = 5 (opposite angle…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle** Vertices: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with: - AB along the x-axis from (0,0) to (4,0), length = 4 - BC vertical from (4,0) to (4,3), length = 3 - AC is the hypotenuse from (0,0) to (4,3), length = sqrt( (4-0)^2 + (3-0)^2 ) = sqrt(16 + 9) = sqrt(25) = 5 So sides: a = BC = 3 (opposite angle A) b = AC = 5 (opposite angle B) c = AB = 4 (opposite angle C) Right angle at B = (4,0). --- **Step 2: Formula for inradius of a triangle** For any triangle: Area = r * s where r = inradius, s = semiperimeter = (a+b+c)/2 So r = Area / s --- **Step 3: Compute area** Right triangle with legs 3 and 4: Area = (1/2) * base * height = (1/2) * 4 * 3 = 6 --- **Step 4: Compute semiperimeter** s = (3 + 4 + 5) / 2 = 12 / 2 = 6 --- **Step 5: Compute inradius** r = Area / s = 6 / 6 = 1 --- **Step 6: Conclusion** The radius of the inscribed circle is 1. --- **Final answer:** 1

5. Emma is studying the population growth of a rare species of orchid in a protected forest. The number of orchids, N(t), after t years is modeled by the exponential function N(t) = 45 × 3^(t/7). Emma wants to determine how many years it will take for the orchid population to reach 1,215 plants. Express your answer in logarithmic form. Answer: t = 7 × log₃(27) or t = 7 × (log(27)/log(3)) Solution: Set up the equation: 45 × 3^(t/7) = 1215 Divide both sides by 45: 3^(t/7) = 1215/45 = 27 Take the logarithm base 3 of both sides: log₃(3^(t/7)) = log₃(27) Apply the power rule of logarithms: (t/7) × log₃(3) = log₃(27) Since log₃(3) = 1, we have: t/7 = log₃(27) Multiply both sides by 7: t = 7 ×…
   Full step-by-step solution

   Step 1: Set up the equation: 45 × 3^(t/7) = 1215 Step 2: Divide both sides by 45: 3^(t/7) = 1215/45 = 27 Step 3: Take the logarithm base 3 of both sides: log₃(3^(t/7)) = log₃(27) Step 4: Apply the power rule of logarithms: (t/7) × log₃(3) = log₃(27) Step 5: Since log₃(3) = 1, we have: t/7 = log₃(27) Step 6: Multiply both sides by 7: t = 7 × log₃(27) Step 7: Using the change of base formula, this can also be written as t = 7 × (log(27)/log(3)) The answer is t = 7 × log₃(27) or equivalently t = 7 × (log(27)/log(3)).

6. A financial analyst is modeling compound interest for an investment account. The account balance A(t) after t years is given by A(t) = 2500 × (1.045)^t. If the investor wants to know when the account will reach $5,000, express the time t in logarithmic form. Answer: t = log(2) / log(1.045) Solution: Set up the equation: 5000 = 2500 × (1.045)^t Divide both sides by 2500: 5000/2500 = (1.045)^t Simplify: 2 = (1.045)^t Apply logarithm to both sides: log(2) = log((1.045)^t) Use the power rule of logarithms: log(2) = t × log(1.045) Solve for t: t = log(2) / log(1.045) The answer is t = log(2) /…
   Full step-by-step solution

   Step 1: Set up the equation: 5000 = 2500 × (1.045)^t Step 2: Divide both sides by 2500: 5000/2500 = (1.045)^t Step 3: Simplify: 2 = (1.045)^t Step 4: Apply logarithm to both sides: log(2) = log((1.045)^t) Step 5: Use the power rule of logarithms: log(2) = t × log(1.045) Step 6: Solve for t: t = log(2) / log(1.045) The answer is t = log(2) / log(1.045).

7. Olivia is an environmental scientist studying the pH of a local river. The hydrogen ion concentration [H⁺] in moles per liter is modeled by the equation [H⁺] = 2.5 × 10^(-5) × (1.20)^t, where t is the number of years since 2020. Olivia wants to know when the hydrogen ion concentration will reach 1.0 × 10^(-4) moles per liter. Express your answer in logarithmic form. Answer: t = log(4) / log(1.20) Solution: Set up the equation for the target concentration: 2.5 × 10^(-5) × (1.20)^t = 1.0 × 10^(-4). Divide both sides by 2.5 × 10^(-5): (1.20)^t = (1.0 × 10^(-4)) / (2.5 × 10^(-5)).
   Full step-by-step solution

   Step 1: Set up the equation for the target concentration: 2.5 × 10^(-5) × (1.20)^t = 1.0 × 10^(-4). Step 2: Divide both sides by 2.5 × 10^(-5): (1.20)^t = (1.0 × 10^(-4)) / (2.5 × 10^(-5)). Step 3: Simplify the right side: (1.0 / 2.5) × (10^(-4) / 10^(-5)) = 0.4 × 10^(1) = 4. Step 4: So the equation is (1.20)^t = 4. Step 5: Take the logarithm of both sides: log((1.20)^t) = log(4). Step 6: Apply the power rule of logarithms: t × log(1.20) = log(4). Step 7: Solve for t: t = log(4) / log(1.20). The answer in logarithmic form is t = log(4) / log(1.20).