Logarithmic Form Solutions

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Logarithmic Form Solutions · Grade 11 · Worksheet 1

1. Liam is analyzing the decay of a radioactive isotope used in medical imaging. The amount A(t) in milligrams after t years is given by A(t) = 200 × (1/2)^(t/12). How many years will it take for the isotope to decay to 25 milligrams? Express your answer in logarithmic form. Answer: t = 12 × (log(200/25)/log(2)) Solution: Start with the decay equation: A(t) = 200 × (1/2)^(t/12) Set A(t) = 25: 25 = 200 × (1/2)^(t/12) Divide both sides by 200: 25/200 = (1/2)^(t/12) Simplify: 1/8 = (1/2)^(t/12) Take logarithm of both sides: log(1/8) = log((1/2)^(t/12)) Use power rule: log(1/8) = (t/12) × log(1/2) Solve for t: t = 12…
   Full step-by-step solution

   Step 1: Start with the decay equation: A(t) = 200 × (1/2)^(t/12) Step 2: Set A(t) = 25: 25 = 200 × (1/2)^(t/12) Step 3: Divide both sides by 200: 25/200 = (1/2)^(t/12) Step 4: Simplify: 1/8 = (1/2)^(t/12) Step 5: Take logarithm of both sides: log(1/8) = log((1/2)^(t/12)) Step 6: Use power rule: log(1/8) = (t/12) × log(1/2) Step 7: Solve for t: t = 12 × (log(1/8)/log(1/2)) Step 8: Simplify using change of base: t = 12 × (log(200/25)/log(2)) The answer is t = 12 × (log(200/25)/log(2))

2. 14^x = 83. Express solution using logarithms. Answer: x = log₁₄(83) Solution: Start with the exponential equation: 14^x = 83 Recall the definition of a logarithm: if a^b = c, then b = logₐ(c), where a > 0, a ≠ 1, and c > 0. Apply this definition to the given equation.
   Full step-by-step solution

   Step 1: Start with the exponential equation: 14^x = 83 Step 2: Recall the definition of a logarithm: if a^b = c, then b = logₐ(c), where a > 0, a ≠ 1, and c > 0. Step 3: Apply this definition to the given equation. Here, a = 14, b = x, and c = 83. Step 4: Therefore, x = log₁₄(83). The solution expressed in logarithmic form is x = log₁₄(83).

3. A biologist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and it grows at a continuous rate of 8% per hour. The biologist needs to determine how many hours it will take for the population to reach 2,000 bacteria. Write your answer in logarithmic form. Answer: t = ln(4) / 0.08 Solution: We are told the initial population is 500 bacteria and it grows at a continuous rate of 8% per hour. We want the time t for the population to reach 2000 bacteria.
   Full step-by-step solution

   We are told the initial population is 500 bacteria and it grows at a continuous rate of 8% per hour. We want the time t for the population to reach 2000 bacteria. --- **Step 1: Set up the continuous growth formula** For continuous growth at rate r (as a decimal), the population at time t is: P(t) = P0 * e^(r t) Here: P0 = 500 r = 0.08 P(t) = 2000 So: 2000 = 500 * e^(0.08 t) --- **Step 2: Divide both sides by 500** 2000 / 500 = 4 So: 4 = e^(0.08 t) --- **Step 3: Take the natural logarithm of both sides** ln(4) = ln(e^(0.08 t)) Since ln(e^x) = x: ln(4) = 0.08 t --- **Step 4: Solve for t** t = ln(4) / 0.08 --- **Final answer:** t = ln(4) / 0.08

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. Lena is studying radioactive decay in her physics lab. She has a sample of Iodine-131 that starts with 800 grams and decays according to the formula A(t) = 800 × (1/2)^(t/8), where t is time in days. Lena needs to determine how many days it will take for the sample to decay to 100 grams. Express your answer in logarithmic form. Answer: t = 24 Solution: Set up the equation using the decay formula: 100 = 800 × (1/2)^(t/8) Divide both sides by 800: 100/800 = (1/2)^(t/8) Simplify the fraction: 1/8 = (1/2)^(t/8) Recognize that 1/8 = (1/2)^3, so: (1/2)^3 = (1/2)^(t/8) Since the bases are equal, set the exponents equal: 3 = t/8 Multiply both sides by…
   Full step-by-step solution

   Step 1: Set up the equation using the decay formula: 100 = 800 × (1/2)^(t/8) Step 2: Divide both sides by 800: 100/800 = (1/2)^(t/8) Step 3: Simplify the fraction: 1/8 = (1/2)^(t/8) Step 4: Recognize that 1/8 = (1/2)^3, so: (1/2)^3 = (1/2)^(t/8) Step 5: Since the bases are equal, set the exponents equal: 3 = t/8 Step 6: Multiply both sides by 8: t = 24 The answer is t = 24 days.

6. A biologist 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 50,000? Express your answer in logarithmic form. Answer: t = (ln(10))/(0.15) Solution: P(t) = 5000 × e^(0.15t) We want the time t when P(t) = 50000. Set up the equation. 50000 = 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) = 50000. Step 1: Set up the equation. 50000 = 5000 × e^(0.15t) Step 2: Divide both sides by 5000 to isolate the exponential term. 50000 / 5000 = e^(0.15t) 10 = e^(0.15t) Step 3: To solve for t, take the natural logarithm (ln) of both sides. ln(10) = ln(e^(0.15t)) Step 4: Use the property ln(e^x) = x. ln(10) = 0.15t Step 5: Solve for t by dividing both sides by 0.15. t = ln(10) / 0.15 This is the exact answer in logarithmic form.

7. A research scientist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and it grows at a continuous rate of 8% per hour. The scientist needs to determine how many hours it will take for the bacterial population to reach 2,000. Express your answer in logarithmic form. Answer: t = ln(4)/0.08 Solution: P(t) = P0 * e^(r*t) Initial population P0 = 500 Growth rate r = 8% per hour = 0.08 (in decimal) Final population P(t) = 2000 2000 = 500 * e^(0.08*t) Divide both sides by 500 to isolate the exponential term.
   Full step-by-step solution

   We start with the continuous growth formula: P(t) = P0 * e^(r*t) Given: Initial population P0 = 500 Growth rate r = 8% per hour = 0.08 (in decimal) Final population P(t) = 2000 Substitute into the equation: 2000 = 500 * e^(0.08*t) Step 1: Divide both sides by 500 to isolate the exponential term. 2000 / 500 = e^(0.08*t) 4 = e^(0.08*t) Step 2: Take the natural logarithm (ln) of both sides to solve for t. ln(4) = ln(e^(0.08*t)) Step 3: Use the property ln(e^x) = x. ln(4) = 0.08 * t Step 4: Solve for t by dividing both sides by 0.08. t = ln(4) / 0.08 This is the exact logarithmic form for the time it takes for the population to grow from 500 to 2000 bacteria.