Logarithm Properties

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Logarithm Properties · Grade 11 · Worksheet 2

1. Matiu is a volcanologist studying the energy released by two volcanic eruptions. He uses the Richter scale for volcanic events, where the magnitude M is given by M = log(E / E_0), with E being the energy released in joules and E_0 being a reference energy of 10^4 joules. The first eruption released 3.2 × 10^15 joules of energy. The second eruption released 6.4 × 10^14 joules of energy. Using the properties of logarithms, how many times more energy did the first eruption release compared to the second? Express your answer as a simplified integer or fraction. Answer: 5 Solution: Write the energies in scientific notation. First eruption: E1 = 3.2 × 10^15 joules. Second eruption: E2 = 6.4 × 10^14 joules.
   Full step-by-step solution

   Step 1: Write the energies in scientific notation. First eruption: E1 = 3.2 × 10^15 joules. Second eruption: E2 = 6.4 × 10^14 joules. Step 2: Find the ratio of E1 to E2 by dividing: E1 / E2 = (3.2 × 10^15) / (6.4 × 10^14). Step 3: Simplify the ratio: Divide the coefficients: 3.2 / 6.4 = 0.5. Divide the powers of 10: 10^15 / 10^14 = 10^(15-14) = 10^1 = 10. So the ratio is 0.5 × 10 = 5. Step 4: Therefore, E1 = 5 × E2, meaning the first eruption released 5 times more energy than the second. The answer is 5.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (8,6). A circle is circumscribed around this triangle, passing through all three vertices. What is the exact area of this circumscribed circle? Answer: 25π Solution: In a right triangle, the hypotenuse is the diameter of the circumscribed circle. Find the hypotenuse length using the Pythagorean theorem: sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 The hypotenuse is the diameter, so the radius is half of 10, which is 5 Calculate the area using the formula…
   Full step-by-step solution

   Step 1: In a right triangle, the hypotenuse is the diameter of the circumscribed circle. Step 2: Find the hypotenuse length using the Pythagorean theorem: sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 Step 3: The hypotenuse is the diameter, so the radius is half of 10, which is 5 Step 4: Calculate the area using the formula A = πr^2 = π(5)^2 = 25π The answer is 25π.

3. log₈(512) + log₉(729) - log₇(343) = ? Answer: 3 Solution: Evaluate log₈(512) 8^x = 512 8^3 = 512 So log₈(512) = 3 Evaluate log₉(729) 9^x = 729 9^3 = 729 So log₉(729) = 3 Evaluate log₇(343) 7^x = 343 7^3 = 343 So log₇(343) = 3 Substitute the values into the original expression 3 + 3 - 3 = 3 The answer is 3.
   Full step-by-step solution

   Step 1: Evaluate log₈(512) 8^x = 512 8^3 = 512 So log₈(512) = 3 Step 2: Evaluate log₉(729) 9^x = 729 9^3 = 729 So log₉(729) = 3 Step 3: Evaluate log₇(343) 7^x = 343 7^3 = 343 So log₇(343) = 3 Step 4: Substitute the values into the original expression 3 + 3 - 3 = 3 The answer is 3.

4. log₃(81x) + log₃(x/9) = 8 Answer: x = 27 Solution: Simplify inside the logarithm: (81x)(x/9) = (81/9)(x·x) = 9x². The equation becomes log₃(9x²) = 8. Rewrite in exponential form: 3⁸ = 9x².
   Full step-by-step solution

   Step 1: Apply the product property: log₃(81x) + log₃(x/9) = log₃((81x)(x/9)). Step 2: Simplify inside the logarithm: (81x)(x/9) = (81/9)(x·x) = 9x². Step 3: The equation becomes log₃(9x²) = 8. Step 4: Rewrite in exponential form: 3⁸ = 9x². Step 5: Compute 3⁸ = 6561, so 6561 = 9x². Step 6: Divide both sides by 9: x² = 729. Step 7: Take the square root: x = √729 = 27 (since x > 0 for the original logarithms to be defined). The answer is x = 27.

5. log₃(27x) + log₃(x/9) = 7 Answer: x = 81 Solution: Simplify inside the logarithm: (27x)(x/9) = 27x²/9 = 3x². Rewrite in exponential form: 3⁷ = 3x². Compute 3⁷ = 2187, so 2187 = 3x².
   Full step-by-step solution

   Step 1: Apply the product rule: log₃(27x) + log₃(x/9) = log₃((27x)(x/9)). Step 2: Simplify inside the logarithm: (27x)(x/9) = 27x²/9 = 3x². So the equation becomes log₃(3x²) = 7. Step 3: Rewrite in exponential form: 3⁷ = 3x². Step 4: Compute 3⁷ = 2187, so 2187 = 3x². Step 5: Divide both sides by 3: x² = 729. Step 6: Take the square root: x = √729 = 27. Step 7: Check for extraneous solutions: The original logarithms require 27x > 0 and x/9 > 0, so x > 0. x = 27 is valid. The answer is x = 27.

6. Dr. Chen 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). She needs to determine how long it will take for the bacterial population to triple from its initial size. Write your answer as an exact logarithmic expression. Answer: (ln(3))/(0.15) Solution: The initial population is when t = 0: P(0) = 500 * e^(0.15 * 0) = 500 * e^0 = 500 * 1 = 500. We want the time t when the population triples, so: P(t) = 3 * 500 = 1500. From P(t) = 500 * e^(0.15t), 1500 = 500 * e^(0.15t).
   Full step-by-step solution

   Step 1: Understand the problem The initial population is when t = 0: P(0) = 500 * e^(0.15 * 0) = 500 * e^0 = 500 * 1 = 500. We want the time t when the population triples, so: P(t) = 3 * 500 = 1500. Step 2: Set up the equation From P(t) = 500 * e^(0.15t), 1500 = 500 * e^(0.15t). Step 3: Isolate the exponential term Divide both sides by 500: 1500 / 500 = e^(0.15t) 3 = e^(0.15t). Step 4: Apply the natural logarithm Take ln of both sides: ln(3) = ln(e^(0.15t)). Step 5: Simplify using logarithm properties ln(e^(0.15t)) = 0.15t * ln(e) = 0.15t * 1 = 0.15t. So: ln(3) = 0.15t. Step 6: Solve for t Divide both sides by 0.15: t = ln(3) / 0.15. Final answer: t = (ln(3))/(0.15)

7. Dr. Chen is studying the decay of a radioactive isotope in a medical sample. The amount of the isotope remaining after t hours is given by A(t) = 500 × e^(-0.023t) milligrams. If the hospital needs at least 200 mg of the isotope for a diagnostic procedure, how many hours can they wait before the sample becomes insufficient? Round your answer to the nearest hour. Answer: 40 Solution: A(t) = 500 × e^(-0.023t) We need to find t when A(t) = 200. Set up the equation. 500 × e^(-0.023t) = 200 Divide both sides by 500.
   Full step-by-step solution

   We are given the decay function: A(t) = 500 × e^(-0.023t) We need to find t when A(t) = 200. Step 1: Set up the equation. 500 × e^(-0.023t) = 200 Step 2: Divide both sides by 500. e^(-0.023t) = 200/500 e^(-0.023t) = 2/5 e^(-0.023t) = 0.4 Step 3: Take the natural logarithm of both sides. ln(e^(-0.023t)) = ln(0.4) -0.023t = ln(0.4) Step 4: Calculate ln(0.4). ln(0.4) ≈ -0.9162907319 Step 5: Substitute into the equation. -0.023t ≈ -0.9162907319 Step 6: Solve for t. t ≈ (-0.9162907319) / (-0.023) t ≈ 39.83872747 Step 7: Round to the nearest hour. t ≈ 40 hours Final answer: 40 hours.