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Logarithms Solve Exponential

Grade 11 · Algebra · Worksheet 2

  1. 7^(x - 3) = 42 Answer: ______________
  2. 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: ______________
  3. Isabella is monitoring the growth of a rare orchid species in a botanical garden. The height of the orchid, in centimeters, is modeled by the function H(t) = 7 × 2^(0.2t), where t is the number of weeks since Isabella began her observation. The orchid will be ready for public display when its height reaches 112 centimeters. How many weeks will it take for the orchid to reach this height? Use logarithms to solve. Answer: ______________
  4. 4^(2x) = 48 Answer: ______________
  5. log₃(4x - 7) = 4 Answer: ______________
  6. 8^(x - 2) = 96 Answer: ______________
  7. 9^(x - 1) = 27 Answer: ______________
  8. 7^(x+2) = 343 Answer: ______________
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Answer Key & Explanations

Logarithms Solve Exponential · Grade 11 · Worksheet 2

  1. 7^(x - 3) = 42 Answer: x ≈ 4.922 Solution: Take the natural logarithm of both sides: ln(7^(x - 3)) = ln(42) Apply the power rule: (x - 3) * ln(7) = ln(42) Divide both sides by ln(7): x - 3 = ln(42) / ln(7) Calculate the logarithms: ln(42) ≈ 3.737, ln(7) ≈ 1.946 Divide: 3.737 / 1.946 ≈ 1.920 Add 3 to both sides: x ≈ 1.920 + 3 = 4.920…
    Full step-by-step solution

    Step 1: Take the natural logarithm of both sides: ln(7^(x - 3)) = ln(42) Step 2: Apply the power rule: (x - 3) * ln(7) = ln(42) Step 3: Divide both sides by ln(7): x - 3 = ln(42) / ln(7) Step 4: Calculate the logarithms: ln(42) ≈ 3.737, ln(7) ≈ 1.946 Step 5: Divide: 3.737 / 1.946 ≈ 1.920 Step 6: Add 3 to both sides: x ≈ 1.920 + 3 = 4.920 Step 7: Round to three decimal places: x ≈ 4.922 The answer is x ≈ 4.922.

  2. 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.

  3. Isabella is monitoring the growth of a rare orchid species in a botanical garden. The height of the orchid, in centimeters, is modeled by the function H(t) = 7 × 2^(0.2t), where t is the number of weeks since Isabella began her observation. The orchid will be ready for public display when its height reaches 112 centimeters. How many weeks will it take for the orchid to reach this height? Use logarithms to solve. Answer: 20 Solution: Set up the equation: 7 × 2^(0.2t) = 112 Divide both sides by 7 to isolate the exponential: 2^(0.2t) = 112 / 7 = 16 Take the common logarithm (log) of both sides: log(2^(0.2t)) = log(16) Apply the power rule of logarithms: 0.2t × log(2) = log(16) Recognize that 16 = 2^4, so log(16) = log(2^4) = 4…
    Full step-by-step solution

    Step 1: Set up the equation: 7 × 2^(0.2t) = 112 Step 2: Divide both sides by 7 to isolate the exponential: 2^(0.2t) = 112 / 7 = 16 Step 3: Take the common logarithm (log) of both sides: log(2^(0.2t)) = log(16) Step 4: Apply the power rule of logarithms: 0.2t × log(2) = log(16) Step 5: Recognize that 16 = 2^4, so log(16) = log(2^4) = 4 × log(2) Step 6: Substitute: 0.2t × log(2) = 4 × log(2) Step 7: Divide both sides by log(2) (which is positive): 0.2t = 4 Step 8: Divide both sides by 0.2: t = 4 / 0.2 = 20 The answer is 20 weeks.

  4. 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.

  5. log₃(4x - 7) = 4 Answer: 22 Solution: Convert the logarithmic equation to exponential form: 3^4 = 4x - 7 Calculate 3^4: 3 × 3 × 3 × 3 = 81 Substitute back: 81 = 4x - 7 Add 7 to both sides: 81 + 7 = 4x → 88 = 4x Divide both sides by 4: 88 ÷ 4 = x → x = 22 The answer is 22.
    Full step-by-step solution

    Step 1: Convert the logarithmic equation to exponential form: 3^4 = 4x - 7 Step 2: Calculate 3^4: 3 × 3 × 3 × 3 = 81 Step 3: Substitute back: 81 = 4x - 7 Step 4: Add 7 to both sides: 81 + 7 = 4x → 88 = 4x Step 5: Divide both sides by 4: 88 ÷ 4 = x → x = 22 The answer is 22.

  6. 8^(x - 2) = 96 Answer: x = (log(96) / log(8)) + 2 ≈ 4.195 Solution: Take the common logarithm (base 10) of both sides: log(8^(x - 2)) = log(96). Use the power rule of logarithms: (x - 2) * log(8) = log(96). Divide both sides by log(8): x - 2 = log(96) / log(8).
    Full step-by-step solution

    Step 1: Take the common logarithm (base 10) of both sides: log(8^(x - 2)) = log(96). Step 2: Use the power rule of logarithms: (x - 2) * log(8) = log(96). Step 3: Divide both sides by log(8): x - 2 = log(96) / log(8). Step 4: Add 2 to both sides: x = (log(96) / log(8)) + 2. Step 5: Approximate using a calculator: log(96) ≈ 1.9823, log(8) ≈ 0.9031, so 1.9823 / 0.9031 ≈ 2.195, then x ≈ 2.195 + 2 = 4.195. The answer is x ≈ 4.195.

  7. 9^(x - 1) = 27 Answer: 2.5 Solution: Express both sides as powers of 3. 9 = 3^2 and 27 = 3^3. Rewrite the equation: (3^2)^(x - 1) = 3^3.
    Full step-by-step solution

    Step 1: Express both sides as powers of 3. 9 = 3^2 and 27 = 3^3. Step 2: Rewrite the equation: (3^2)^(x - 1) = 3^3. Step 3: Apply the power rule: 3^(2(x - 1)) = 3^3. Step 4: Since the bases are equal, set the exponents equal: 2(x - 1) = 3. Step 5: Expand: 2x - 2 = 3. Step 6: Add 2 to both sides: 2x = 5. Step 7: Divide by 2: x = 5/2 = 2.5. The answer is 2.5.

  8. 7^(x+2) = 343 Answer: 1 Solution: Recognize that 343 = 7^3. Step 2: Rewrite the equation as 7^(x+2) = 7^3. Step 3: Since the bases are the same, set the exponents equal: x + 2 = 3.
    Full step-by-step solution

    Step 1: Recognize that 343 = 7^3. Step 2: Rewrite the equation as 7^(x+2) = 7^3. Step 3: Since the bases are the same, set the exponents equal: x + 2 = 3. Step 4: Subtract 2 from both sides: x = 1. The answer is 1.