Logarithm Properties

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Logarithm Properties · Grade 11 · Worksheet 2

1. 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.

2. Aroha is an environmental scientist studying the acidity of rainwater in a forest. She measures the pH of a sample using the formula pH = -log[H⁺], where [H⁺] is the hydrogen ion concentration in moles per liter. She collects two samples: one from an area affected by pollution with a hydrogen ion concentration of 7 × 10⁻⁵ M, and another from a pristine area with a concentration of 1 × 10⁻⁶ M. Using properties of logarithms, determine how many times more acidic the polluted sample is compared to the pristine sample. Answer: 70 Solution: The acidity comparison is based on the ratio of hydrogen ion concentrations: [H⁺]_polluted / [H⁺]_pristine. Substitute the values: (7 × 10⁻⁵) / (1 × 10⁻⁶). Simplify the ratio: 7 × 10⁻⁵ ÷ 10⁻⁶ = 7 × 10⁻⁵⁺⁶ = 7 × 10¹ = 70.
   Full step-by-step solution

   Step 1: The acidity comparison is based on the ratio of hydrogen ion concentrations: [H⁺]_polluted / [H⁺]_pristine. Step 2: Substitute the values: (7 × 10⁻⁵) / (1 × 10⁻⁶). Step 3: Simplify the ratio: 7 × 10⁻⁵ ÷ 10⁻⁶ = 7 × 10⁻⁵⁺⁶ = 7 × 10¹ = 70. Step 4: Therefore, the polluted sample is 70 times more acidic than the pristine sample. The answer is 70.

3. A right triangle is positioned 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: Identify the triangle's sides. The vertices are (0,0), (8,0), and (8,6), so the legs are 8 units and 6 units.
   Full step-by-step solution

   Step 1: Identify the triangle's sides. The vertices are (0,0), (8,0), and (8,6), so the legs are 8 units and 6 units. Step 2: Calculate the hypotenuse using the Pythagorean theorem: hypotenuse = sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 units. Step 3: For any right triangle, the hypotenuse is the diameter of the circumscribed circle. Therefore, the circle's diameter is 10 units. Step 4: Calculate the radius: radius = diameter/2 = 10/2 = 5 units. Step 5: Calculate the area of the circle: area = π × radius^2 = π × 5^2 = 25π. The exact area is 25π.

4. Hana is a materials scientist studying the acidity of industrial wastewater. She measures the pH of two samples using the formula pH = -log[H⁺], where [H⁺] is the hydrogen ion concentration in moles per liter. The first sample has a hydrogen ion concentration of 8.0 × 10⁻⁶ M, and the second sample has a hydrogen ion concentration of 2.0 × 10⁻⁸ M. Using properties of logarithms, determine how many times more acidic the first sample is compared to the second sample. Answer: 400 Solution: The acidity comparison is the ratio of hydrogen ion concentrations: [H⁺]₁ / [H⁺]₂ = (8.0 × 10⁻⁶) / (2.0 × 10⁻⁸). Step 2: Divide the coefficients: 8.0 / 2.0 = 4.
   Full step-by-step solution

   Step 1: The acidity comparison is the ratio of hydrogen ion concentrations: [H⁺]₁ / [H⁺]₂ = (8.0 × 10⁻⁶) / (2.0 × 10⁻⁸). Step 2: Divide the coefficients: 8.0 / 2.0 = 4. Step 3: Divide the powers of ten: 10⁻⁶ / 10⁻⁸ = 10⁻⁶⁻⁽⁻⁸⁾ = 10². Step 4: Multiply: 4 × 10² = 400. Step 5: The first sample is 400 times more acidic than the second sample. The answer is 400.

5. log₃(x) + log₃(x - 8) = 2 Answer: 9 Solution: Rewrite in exponential form: 3² = x(x - 8). Simplify: 9 = x² - 8x. Rearrange to standard form: x² - 8x - 9 = 0.
   Full step-by-step solution

   Step 1: Apply the product property: log₃(x) + log₃(x - 8) = log₃[x(x - 8)] = 2. Step 2: Rewrite in exponential form: 3² = x(x - 8). Step 3: Simplify: 9 = x² - 8x. Step 4: Rearrange to standard form: x² - 8x - 9 = 0. Step 5: Factor the quadratic: (x - 9)(x + 1) = 0. Step 6: Solve: x = 9 or x = -1. Step 7: Check for extraneous solutions: For x = 9, log₃(9) and log₃(1) are defined (both arguments positive). For x = -1, log₃(-1) is undefined. So x = -1 is extraneous. The answer is 9.

6. Ava is an environmental scientist studying the pH levels of rainwater in a polluted region. She measures the hydrogen ion concentration [H⁺] of a sample to be 6.31 × 10⁻⁶ moles per liter. The pH is defined by the formula pH = -log[H⁺]. Using the properties of logarithms, determine the pH of the rainwater sample. Round your final answer to one decimal place. Answer: 5.2 Solution: Write the pH formula: pH = -log[H⁺] Substitute the given concentration: pH = -log(6.31 × 10⁻⁶) Apply the product property of logarithms: log(6.31 × 10⁻⁶) = log(6.31) + log(10⁻⁶) Evaluate log(10⁻⁶) = -6 (since log base 10 of 10 to any power is that power) Use a calculator to find log(6.31) ≈…
   Full step-by-step solution

   Step 1: Write the pH formula: pH = -log[H⁺] Step 2: Substitute the given concentration: pH = -log(6.31 × 10⁻⁶) Step 3: Apply the product property of logarithms: log(6.31 × 10⁻⁶) = log(6.31) + log(10⁻⁶) Step 4: Evaluate log(10⁻⁶) = -6 (since log base 10 of 10 to any power is that power) Step 5: Use a calculator to find log(6.31) ≈ 0.8000 Step 6: So, log(6.31 × 10⁻⁶) = 0.8000 + (-6) = -5.2000 Step 7: Now apply the negative sign: pH = -(-5.2000) = 5.2000 Step 8: Round to one decimal place: pH = 5.2 The answer is 5.2.

7. Noah is analyzing the graph of a logarithmic function f(x) = log_7(49x^3). Using properties of logarithms, rewrite f(x) as a sum of simpler logarithmic terms, and then determine the value of f(7) exactly. Answer: 5 Solution: Evaluate log_7(49). Since 49 = 7^2, we have log_7(49) = 2. So f(x) = 2 + 3 * log_7(x).
   Full step-by-step solution

   Step 1: Apply the product rule to separate 49 and x^3: log_7(49x^3) = log_7(49) + log_7(x^3). Step 2: Evaluate log_7(49). Since 49 = 7^2, we have log_7(49) = 2. Step 3: Apply the power rule to log_7(x^3): log_7(x^3) = 3 * log_7(x). Step 4: So f(x) = 2 + 3 * log_7(x). Step 5: To find f(7), substitute x = 7: f(7) = 2 + 3 * log_7(7). Step 6: Since log_7(7) = 1, we get f(7) = 2 + 3 * 1 = 5. The answer is 5.