Logarithms Solve Exponential

lessonbunny.com

Answer Key & Explanations

Logarithms Solve Exponential · Grade 11 · Worksheet 1

1. Aroha is a conservation biologist studying the recovery of a native bird population on a predator-free island. The number of birds, N(t), after t years is modeled by the function N(t) = 7 × 5^(0.3t). The island's ecosystem can sustainably support a maximum of 875 birds. How many years will it take for the bird population to reach this sustainable capacity? Use logarithms to solve. Answer: 10 Solution: Set up the equation: 7 × 5^(0.3t) = 875 Divide both sides by 7: 5^(0.3t) = 875 / 7 = 125 Take the common logarithm (log) of both sides: log(5^(0.3t)) = log(125) Apply the power rule of logarithms: 0.3t × log(5) = log(125) Recognize that 125 = 5^3, so log(125) = log(5^3) = 3 × log(5) Substitute:…
   Full step-by-step solution

   Step 1: Set up the equation: 7 × 5^(0.3t) = 875 Step 2: Divide both sides by 7: 5^(0.3t) = 875 / 7 = 125 Step 3: Take the common logarithm (log) of both sides: log(5^(0.3t)) = log(125) Step 4: Apply the power rule of logarithms: 0.3t × log(5) = log(125) Step 5: Recognize that 125 = 5^3, so log(125) = log(5^3) = 3 × log(5) Step 6: Substitute: 0.3t × log(5) = 3 × log(5) Step 7: Divide both sides by log(5) (positive): 0.3t = 3 Step 8: Divide both sides by 0.3: t = 3 / 0.3 = 10 The answer is 10 years.

2. 4^(2x) = 48 Answer: x = (log(48))/(2 log(4)) ≈ 1.396 Solution: Take the natural log (or common log) of both sides: ln(4^(2x)) = ln(48). Divide both sides by 2 ln(4): x = ln(48) / (2 ln(4)). Simplify if desired: ln(48) = ln(16 * 3) = ln(16) + ln(3) = 4 ln(2) + ln(3).
   Full step-by-step solution

   Step 1: Take the natural log (or common log) of both sides: ln(4^(2x)) = ln(48). Step 2: Apply the power rule: 2x ln(4) = ln(48). Step 3: Divide both sides by 2 ln(4): x = ln(48) / (2 ln(4)). Step 4: Simplify if desired: ln(48) = ln(16 * 3) = ln(16) + ln(3) = 4 ln(2) + ln(3). Also ln(4) = 2 ln(2). So x = (4 ln(2) + ln(3)) / (4 ln(2)) = 1 + ln(3)/(4 ln(2)). Step 5: Approximate: ln(3) ≈ 1.0986, ln(2) ≈ 0.6931, so x ≈ 1 + 1.0986/(4*0.6931) = 1 + 1.0986/2.7724 ≈ 1 + 0.396 = 1.396. The answer is x = ln(48)/(2 ln(4)) ≈ 1.396.

3. Mere is analyzing the growth of a bacterial colony on a Petri dish. The colony grows exponentially according to the equation A = 15 * 4^(t), where A is the area of the colony in square millimeters after t hours. On a graph of this exponential growth, the point where the colony reaches an area of 240 square millimeters is marked. Using logarithms, determine the value of t when A = 240. Answer: 2 Solution: Write the equation with the given area: 240 = 15 * 4^t Divide both sides by 15: 240/15 = 4^t, so 16 = 4^t Take the logarithm of both sides.
   Full step-by-step solution

   Step 1: Write the equation with the given area: 240 = 15 * 4^t Step 2: Divide both sides by 15: 240/15 = 4^t, so 16 = 4^t Step 3: Take the logarithm of both sides. Use base 10 or natural log: log(16) = log(4^t) Step 4: Apply the power rule of logarithms: log(16) = t * log(4) Step 5: Solve for t: t = log(16) / log(4) Step 6: Calculate: log(16) ≈ 1.20412 and log(4) ≈ 0.60206 Step 7: t = 1.20412 / 0.60206 = 2 The answer is 2.

4. Aroha is monitoring the population of a rare species of tree frog in a protected rainforest. The population, P(t), after t years is modeled by the function P(t) = 9 × 5^(0.15t). Conservationists have set a recovery target of 1,125 frogs. How many years will it take for the frog population to reach this target? Use logarithms to solve. Answer: 20 Solution: Set up the equation: 9 × 5^(0.15t) = 1125. Divide both sides by 9 to isolate the exponential: 5^(0.15t) = 1125 / 9 = 125. Take the common logarithm (log) of both sides: log(5^(0.15t)) = log(125).
   Full step-by-step solution

   Step 1: Set up the equation: 9 × 5^(0.15t) = 1125. Step 2: Divide both sides by 9 to isolate the exponential: 5^(0.15t) = 1125 / 9 = 125. Step 3: Take the common logarithm (log) of both sides: log(5^(0.15t)) = log(125). Step 4: Apply the power rule of logarithms: 0.15t × log(5) = log(125). Step 5: Recognize that 125 = 5^3, so log(125) = log(5^3) = 3 × log(5). Step 6: Substitute: 0.15t × log(5) = 3 × log(5). Step 7: Divide both sides by log(5) (which is positive): 0.15t = 3. Step 8: Divide both sides by 0.15: t = 3 / 0.15 = 20. The answer is 20 years.

5. log₃(x) + log₃(x + 6) = 3 Answer: 3 Solution: Step 1: Apply the product rule for logarithms: log₃(x) + log₃(x + 6) = log₃(x(x + 6)) Step 2: The equation becomes: log₃(x(x + 6)) = 3 Step 3: Convert to exponential form: x(x + 6) = 3³ Step 4: Simplify: x(x + 6) = 27 Step 5: Expand: x² + 6x = 27 Step 6: Rearrange: x² + 6x - 27 = 0 Step 7:…
   Full step-by-step solution

   Step 1: Apply the product rule for logarithms: log₃(x) + log₃(x + 6) = log₃(x(x + 6)) Step 2: The equation becomes: log₃(x(x + 6)) = 3 Step 3: Convert to exponential form: x(x + 6) = 3³ Step 4: Simplify: x(x + 6) = 27 Step 5: Expand: x² + 6x = 27 Step 6: Rearrange: x² + 6x - 27 = 0 Step 7: Factor: (x + 9)(x - 3) = 0 Step 8: Solve: x = -9 or x = 3 Step 9: Check domain: log₃(x) requires x > 0, so x = -9 is extraneous Step 10: The valid solution is x = 3

6. Aroha is studying a bacterial culture that grows exponentially according to the model P(t) = P₀ * 3^(t/5), where P₀ is the initial population and t is time in hours. She graphs the population on a coordinate plane with time on the x-axis and population on the y-axis. After some time, the population reaches 729 times its initial value. Determine the time t (in hours) when this occurs using logarithms. Answer: 30 Solution: Write the equation when P(t) = 729 * P₀. P₀ * 3^(t/5) = 729 * P₀ Divide both sides by P₀. 3^(t/5) = 729 Express 729 as a power of 3.
   Full step-by-step solution

   Step 1: Write the equation when P(t) = 729 * P₀. P₀ * 3^(t/5) = 729 * P₀ Step 2: Divide both sides by P₀. 3^(t/5) = 729 Step 3: Express 729 as a power of 3. 729 = 3^6 because 3^6 = 3*3*3*3*3*3 = 729. Step 4: Set the exponents equal since the bases are the same. t/5 = 6 Step 5: Multiply both sides by 5. t = 30 The answer is 30.

7. A biologist is studying bacterial growth in a lab culture. The population P(t) follows the exponential model P(t) = 5000 × e^(0.03t), where t is time in hours. If the lab's containment system can only handle 15,000 bacteria safely, how many hours (to the nearest tenth) can pass before the population reaches this safety limit? Answer: 36.6 Solution: P(t) = 5000 × e^(0.03t) We want the time t when P(t) = 15000. Set up the equation. 15000 = 5000 × e^(0.03t) Divide both sides by 5000 to isolate the exponential term.
   Full step-by-step solution

   We are given the exponential growth model: P(t) = 5000 × e^(0.03t) We want the time t when P(t) = 15000. Step 1: Set up the equation. 15000 = 5000 × e^(0.03t) Step 2: Divide both sides by 5000 to isolate the exponential term. 15000 / 5000 = e^(0.03t) 3 = e^(0.03t) Step 3: Take the natural logarithm of both sides to solve for t. ln(3) = ln(e^(0.03t)) ln(3) = 0.03t × ln(e) Since ln(e) = 1, we have: ln(3) = 0.03t Step 4: Solve for t. t = ln(3) / 0.03 Step 5: Calculate numerical values. ln(3) ≈ 1.0986122887 t ≈ 1.0986122887 / 0.03 t ≈ 36.6204096233 Step 6: Round to the nearest tenth. t ≈ 36.6 hours Thus, the population reaches 15000 after about 36.6 hours.