Logarithm Properties

lessonbunny.com

Answer Key & Explanations

Logarithm Properties · Grade 11 · Worksheet 1

1. Mason is studying a logarithmic spiral on a polar coordinate grid. The spiral is described by the equation r = 9 * 3^(θ/2), where r is the distance from the origin and θ is the angle in radians. Using properties of logarithms, rewrite the expression for log₃(r) as a linear function of θ. Then, determine the exact value of log₃(r) when θ = 8 radians. Answer: 6 Solution: Take log base 3 of both sides of r = 9 * 3^(θ/2). This gives log₃(r) = log₃(9 * 3^(θ/2)). Evaluate log₃(9).
   Full step-by-step solution

   Step 1: Take log base 3 of both sides of r = 9 * 3^(θ/2). This gives log₃(r) = log₃(9 * 3^(θ/2)). Step 2: Apply the product rule: log₃(9 * 3^(θ/2)) = log₃(9) + log₃(3^(θ/2)). Step 3: Evaluate log₃(9). Since 9 = 3^2, we have log₃(9) = 2. Step 4: Apply the power rule to log₃(3^(θ/2)): log₃(3^(θ/2)) = (θ/2) * log₃(3). Step 5: Since log₃(3) = 1, this simplifies to θ/2. Step 6: So log₃(r) = 2 + θ/2, which is a linear function of θ. Step 7: Now substitute θ = 8: log₃(r) = 2 + 8/2 = 2 + 4 = 6. The answer is 6.

2. log₂(x) + log₂(x - 7) = 3 Answer: 8 Solution: The equation becomes log₂(x(x - 7)) = 3. Rewrite in exponential form: 2³ = x(x - 7). Simplify: 8 = x² - 7x.
   Full step-by-step solution

   Step 1: Apply the product property: log₂(x) + log₂(x - 7) = log₂(x(x - 7)). The equation becomes log₂(x(x - 7)) = 3. Step 2: Rewrite in exponential form: 2³ = x(x - 7). Step 3: Simplify: 8 = x² - 7x. Step 4: Rearrange to standard form: x² - 7x - 8 = 0. Step 5: Factor the quadratic: (x - 8)(x + 1) = 0. Step 6: Solve: x = 8 or x = -1. Step 7: Check domain: For log₂(x), x > 0. For log₂(x - 7), x - 7 > 0, so x > 7. x = 8 satisfies both conditions. x = -1 is not in the domain. The answer is 8.

3. A biologist is studying bacterial growth in a lab culture. The population P(t) after t hours is modeled by the exponential function P(t) = 5000 × e^(0.15t). Liam needs to determine how long it will take for the bacterial population to reach 25,000. Using properties of logarithms, find the time required for the population to reach this level. Answer: 10.73 hours Solution: P(t) = 5000 × e^(0.15t) We want to find t when P(t) = 25000. Set up the equation. 25000 = 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 to find t when P(t) = 25000. Step 1: Set up the equation. 25000 = 5000 × e^(0.15t) Step 2: Divide both sides by 5000 to isolate the exponential term. 25000 / 5000 = e^(0.15t) 5 = e^(0.15t) Step 3: Take the natural logarithm of both sides to solve for t. ln(5) = ln(e^(0.15t)) Step 4: Use the property ln(e^x) = x. ln(5) = 0.15t Step 5: Solve for t. t = ln(5) / 0.15 Step 6: Calculate numerical values. ln(5) ≈ 1.609437912 t ≈ 1.609437912 / 0.15 t ≈ 10.72958608 Step 7: Round to two decimal places. t ≈ 10.73 hours Thus, it will take about 10.73 hours for the population to reach 25,000.

4. Tane is analyzing the pH levels of two solutions in chemistry class. The pH of solution A is 3 and the pH of solution B is 1. pH is defined as -log[H⁺]. Using the quotient property of logarithms (log(a/b) = log a - log b), if the hydrogen ion concentration of solution A is divided by that of solution B, what is the resulting pH value? Answer: 2 Solution: pH of solution A = 3, pH of solution B = 1. The quotient property states: log([H⁺]A / [H⁺]B) = log([H⁺]A) - log([H⁺]B). Since pH = -log[H⁺], the difference in pH = 3 - 1 = 2.
   Full step-by-step solution

   Step 1: pH of solution A = 3, pH of solution B = 1. Step 2: The quotient property states: log([H⁺]A / [H⁺]B) = log([H⁺]A) - log([H⁺]B). Step 3: Since pH = -log[H⁺], the difference in pH = 3 - 1 = 2. The answer is 2.

5. Sophia is a chemist analyzing the acidity of two rainwater samples collected from different locations. She uses the pH 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 × 10⁻⁷ M. Using the properties of logarithms, how many times more acidic is the first sample than the second sample? Answer: 40 Solution: Write the concentrations: first sample [H⁺]₁ = 8.0 × 10⁻⁶ M, second sample [H⁺]₂ = 2.0 × 10⁻⁷ M.
   Full step-by-step solution

   Step 1: Write the concentrations: first sample [H⁺]₁ = 8.0 × 10⁻⁶ M, second sample [H⁺]₂ = 2.0 × 10⁻⁷ M. Step 2: To find how many times more acidic the first sample is, compute the ratio: ratio = [H⁺]₁ / [H⁺]₂ = (8.0 × 10⁻⁶) / (2.0 × 10⁻⁷). Step 3: Divide the coefficients: 8.0 / 2.0 = 4.0. Step 4: Divide the powers of 10: 10⁻⁶ / 10⁻⁷ = 10^(-6 - (-7)) = 10^(1) = 10. Step 5: Multiply: 4.0 × 10 = 40. Step 6: Therefore, the first sample is 40 times more acidic than the second sample. The answer is 40.

6. Matiu is a volcanologist studying the intensity of an earthquake near a volcano. The Richter scale magnitude M is given by the formula M = log(I/I₀), where I is the intensity of the earthquake and I₀ is a reference intensity. Matiu records two earthquakes: one with a magnitude of 5.2 and another with a magnitude of 6.4. Using the properties of logarithms, determine how many times more intense the larger earthquake is compared to the smaller one. Round your answer to the nearest whole number. Answer: 16 Solution: Let I₁ be the intensity of the magnitude 5.2 earthquake, and I₂ be the intensity of the magnitude 6.4 earthquake. We have: M₁ = log(I₁/I₀) = 5.2, M₂ = log(I₂/I₀) = 6.4. The ratio of intensities I₂/I₁ is what we want.
   Full step-by-step solution

   Step 1: Let I₁ be the intensity of the magnitude 5.2 earthquake, and I₂ be the intensity of the magnitude 6.4 earthquake. We have: M₁ = log(I₁/I₀) = 5.2, M₂ = log(I₂/I₀) = 6.4. Step 2: The ratio of intensities I₂/I₁ is what we want. Use the difference: M₂ - M₁ = log(I₂/I₀) - log(I₁/I₀). Step 3: Apply the quotient rule of logarithms: log(I₂/I₀) - log(I₁/I₀) = log[(I₂/I₀) ÷ (I₁/I₀)] = log(I₂/I₁). Step 4: Substitute the magnitudes: 6.4 - 5.2 = log(I₂/I₁), so 1.2 = log(I₂/I₁). Step 5: Rewrite in exponential form: I₂/I₁ = 10^1.2. Step 6: Calculate 10^1.2: 10^1.2 = 10^(1 + 0.2) = 10^1 × 10^0.2 = 10 × 1.5849 = 15.849. Step 7: Round to the nearest whole number: 16. The answer is 16.