Logarithmic Form Solutions

lessonbunny.com

Answer Key & Explanations

Logarithmic Form Solutions · Grade 11 · Worksheet 3

1. 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).

2. 11^x = 37. Express solution using logarithms Answer: x = log₁₁(37) Solution: Start with the exponential equation: 11^x = 37 To solve for x, convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c) Apply this to our equation: x = log₁₁(37) The solution in logarithmic form is x = log₁₁(37).
   Full step-by-step solution

   Step 1: Start with the exponential equation: 11^x = 37 Step 2: To solve for x, convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c) Step 3: Apply this to our equation: x = log₁₁(37) The solution in logarithmic form is x = log₁₁(37).

3. 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 such that it is tangent to all three sides. What is the radius of this inscribed circle? Answer: 1 Solution: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0).
   Full step-by-step solution

   Let's solve step by step. --- **Step 1: Understand the triangle** Vertices: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0). Check: AB is horizontal from (0,0) to (4,0) → length = 4 BC is 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 vertex A) b = AC = 5 (opposite vertex B) c = AB = 4 (opposite vertex C) --- **Step 2: Formula for inradius of a right triangle** For any triangle: Area = r × s where r = inradius, s = semiperimeter = (a + b + c)/2 So r = Area / s --- **Step 3: Compute area and semiperimeter** Area = (1/2) × (leg1) × (leg2) = (1/2) × 3 × 4 = 6 Semiperimeter s = (3 + 4 + 5)/2 = 12/2 = 6 --- **Step 4: Compute r** r = Area / s = 6 / 6 = 1 --- **Step 5: Conclusion** The radius of the inscribed circle is 1. --- **Final answer:** 1

4. An archaeologist is analyzing the decay of carbon-14 in an ancient artifact. The artifact originally contained 120 micrograms of carbon-14, but now contains only 15 micrograms. Given that carbon-14 has a half-life of 5,730 years, determine how many years have passed since the artifact was created. Express your answer in logarithmic form. Answer: t = 5730 * ln(8)/ln(2) Solution: Use the exponential decay formula: A = A₀ * (1/2)^(t/h), where A is current amount, A₀ is original amount, t is time, 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 original amount, t is time, h is half-life. Step 2: Substitute 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: Rewrite 1/8 as (1/2)^3: (1/2)^3 = (1/2)^(t/5730) Step 6: Since the bases are equal, set exponents equal: 3 = t/5730 Step 7: Multiply both sides by 5730: t = 3 * 5730 Step 8: Express in logarithmic form using the relationship from step 5: t = 5730 * log₁/₂(1/8) Step 9: Convert to natural logarithms: t = 5730 * ln(1/8)/ln(1/2) Step 10: Simplify using logarithm properties: t = 5730 * [-ln(8)]/[-ln(2)] = 5730 * ln(8)/ln(2) The answer is t = 5730 * ln(8)/ln(2)

5. 3^(2x+1) = 75. Express solution using logarithms Answer: x = (log₃(75) - 1)/2 Solution: Start with the equation: 3^(2x+1) = 75 Convert to logarithmic form: 2x+1 = log₃(75) Subtract 1 from both sides: 2x = log₃(75) - 1 Divide both sides by 2: x = (log₃(75) - 1)/2 The solution expressed in logarithmic form is x = (log₃(75) - 1)/2
   Full step-by-step solution

   Step 1: Start with the equation: 3^(2x+1) = 75 Step 2: Convert to logarithmic form: 2x+1 = log₃(75) Step 3: Subtract 1 from both sides: 2x = log₃(75) - 1 Step 4: Divide both sides by 2: x = (log₃(75) - 1)/2 Step 5: The solution expressed in logarithmic form is x = (log₃(75) - 1)/2

6. 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) = 500 × e^(0.15t). After how many hours will the bacterial population reach 10,000? Express your answer in logarithmic form. Answer: t = (ln(20))/(0.15) Solution: P(t) = 500 × e^(0.15t) We want the time t when P(t) = 10000. Set up the equation. 10000 = 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) We want the time t when P(t) = 10000. Step 1: Set up the equation. 10000 = 500 × e^(0.15t) Step 2: Divide both sides by 500 to isolate the exponential term. 10000 / 500 = e^(0.15t) 20 = e^(0.15t) Step 3: To solve for t, take the natural logarithm (ln) of both sides. ln(20) = ln(e^(0.15t)) Step 4: Use the property ln(e^x) = x. ln(20) = 0.15t Step 5: Solve for t by dividing both sides by 0.15. t = ln(20) / 0.15 This is the exact answer in logarithmic form.

7. 7^x = 50. Express solution using logarithms Answer: x = log₇(50) Solution: Start with the equation 7^x = 50 Apply the definition of logarithms: if a^b = c, then b = logₐ(c) Here, a = 7, b = x, and c = 50 Therefore, x = log₇(50) The solution in logarithmic form is x = log₇(50)
   Full step-by-step solution

   Step 1: Start with the equation 7^x = 50 Step 2: Apply the definition of logarithms: if a^b = c, then b = logₐ(c) Step 3: Here, a = 7, b = x, and c = 50 Step 4: Therefore, x = log₇(50) The solution in logarithmic form is x = log₇(50)

8. A biologist is studying bacterial growth in a lab culture. The population P(t) after t hours is modeled by the equation P(t) = 500 * e^(0.15t). After how many hours will the bacterial population reach 10,000? Express your answer in logarithmic form. Answer: t = (ln(20)) / 0.15 Solution: P(t) = 500 * e^(0.15t) We want the time t when P(t) = 10,000. Set up the equation. 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) We want the time t when P(t) = 10,000. Step 1: Set up the equation. 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: To solve for t, take the natural logarithm (ln) of both sides. ln(20) = ln(e^(0.15t)) Step 4: Use the property ln(e^x) = x. ln(20) = 0.15t Step 5: Solve for t by dividing both sides by 0.15. t = ln(20) / 0.15 This is the exact answer in logarithmic form.