Logarithms Solve Exponential

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Logarithms Solve Exponential · Grade 11 · Worksheet 3

1. Dr. Chen is studying bacterial growth in a lab culture. The number of bacteria doubles every 4 hours. If she starts with 500 bacteria, how many hours will it take for the population to reach 32,000 bacteria? Use logarithms to solve. Answer: 24 Solution: Let’s go step-by-step. The bacteria double every 4 hours. Initial number of bacteria = 500.
   Full step-by-step solution

   Let’s go step-by-step. --- **Step 1: Understand the growth model** The bacteria double every 4 hours. Initial number of bacteria = 500. We want the time \( t \) (in hours) to reach 32,000 bacteria. The formula for exponential growth is: \[ N(t) = N_0 \times 2^{t / d} \] where \( N_0 = 500 \) \( d = 4 \) hours (doubling time) \( N(t) = 32000 \) --- **Step 2: Set up the equation** \[ 32000 = 500 \times 2^{t / 4} \] --- **Step 3: Isolate the power term** Divide both sides by 500: \[ \frac{32000}{500} = 2^{t / 4} \] \[ 64 = 2^{t / 4} \] --- **Step 4: Recognize powers of 2** \( 64 = 2^6 \), so: \[ 2^{t / 4} = 2^6 \] --- **Step 5: Equate exponents** Since the bases are the same (base 2): \[ \frac{t}{4} = 6 \] --- **Step 6: Solve for \( t \)** \[ t = 6 \times 4 = 24 \] --- **Step 7: Conclusion** It will take 24 hours for the bacteria population to grow from 500 to 32,000. --- **Using logarithms (as requested in the problem):** From \( 64 = 2^{t / 4} \), take \(\log\) of both sides: \[ \log(64) = \frac{t}{4} \log(2) \] \[ \frac{t}{4} = \frac{\log(64)}{\log(2)} \] But \(\log(64) / \log(2) = \log_2(64) = 6\), so: \[ \frac{t}{4} = 6 \] \[ t = 24 \] --- **Final answer:** 24 hours

2. 14^(x - 5) = 89 Answer: x ≈ 6.70 Solution: Take the common logarithm (base 10) of both sides: log(14^(x - 5)) = log(89) Apply the power rule: (x - 5) * log(14) = log(89) Divide both sides by log(14): x - 5 = log(89) / log(14) Calculate the logs: log(89) ≈ 1.9494, log(14) ≈ 1.1461 Divide: 1.9494 / 1.1461 ≈ 1.7009 Add 5 to both sides: x ≈…
   Full step-by-step solution

   Step 1: Take the common logarithm (base 10) of both sides: log(14^(x - 5)) = log(89) Step 2: Apply the power rule: (x - 5) * log(14) = log(89) Step 3: Divide both sides by log(14): x - 5 = log(89) / log(14) Step 4: Calculate the logs: log(89) ≈ 1.9494, log(14) ≈ 1.1461 Step 5: Divide: 1.9494 / 1.1461 ≈ 1.7009 Step 6: Add 5 to both sides: x ≈ 1.7009 + 5 = 6.7009 The answer is x ≈ 6.70 (rounded to two decimal places).

3. 5^(2x - 1) = 125^x Answer: x = -1 Solution: Rewrite 125 as a power of 5. Since 125 = 5^3, the equation becomes 5^(2x - 1) = (5^3)^x. Simplify the right side using the power rule: (5^3)^x = 5^(3x).
   Full step-by-step solution

   Step 1: Rewrite 125 as a power of 5. Since 125 = 5^3, the equation becomes 5^(2x - 1) = (5^3)^x. Step 2: Simplify the right side using the power rule: (5^3)^x = 5^(3x). So we have 5^(2x - 1) = 5^(3x). Step 3: Since the bases are equal, set the exponents equal: 2x - 1 = 3x. Step 4: Subtract 2x from both sides: -1 = x. The answer is x = -1.

4. A radioactive substance decays according to the exponential model N(t) = N₀e^(-λt), where N₀ is the initial amount. A scientist observes that after 8 years, only 25% of the original substance remains. The scientist graphs the decay function on a coordinate plane where the x-axis represents time in years and the y-axis represents the percentage of substance remaining. Determine the decay constant λ (to three decimal places) using logarithmic methods. Answer: 0.173 Solution: We are given: N(t) = N0 * e^(-λt) After 8 years, 25% remains, so N(8) = 0.25 * N0. Substitute into the equation. 0.25 * N0 = N0 * e^(-λ * 8) Divide both sides by N0 (N0 > 0).
   Full step-by-step solution

   We are given: N(t) = N0 * e^(-λt) After 8 years, 25% remains, so N(8) = 0.25 * N0. Step 1: Substitute into the equation. 0.25 * N0 = N0 * e^(-λ * 8) Step 2: Divide both sides by N0 (N0 > 0). 0.25 = e^(-8λ) Step 3: Take natural logarithm of both sides. ln(0.25) = ln(e^(-8λ)) ln(0.25) = -8λ * ln(e) Since ln(e) = 1, we have: ln(0.25) = -8λ Step 4: Solve for λ. λ = - ln(0.25) / 8 Step 5: Calculate ln(0.25). 0.25 = 1/4, so ln(0.25) = ln(1) - ln(4) = 0 - ln(4) = -ln(4). ln(4) = ln(2^2) = 2 * ln(2) ≈ 2 * 0.693147 = 1.386294 So ln(0.25) ≈ -1.386294 Step 6: Substitute into λ. λ = - (-1.386294) / 8 λ = 1.386294 / 8 λ ≈ 0.17328675 Step 7: Round to three decimal places. 0.17328675 → 0.173 Final answer: λ ≈ 0.173

5. 7^(2x - 1) = 27 Answer: x = (log₇(27) + 1) / 2 ≈ 1.347 Solution: Take the natural logarithm of both sides: ln(7^(2x - 1)) = ln(27) Use the power rule of logarithms: (2x - 1) * ln(7) = ln(27) Divide both sides by ln(7): 2x - 1 = ln(27) / ln(7) Add 1 to both sides: 2x = ln(27) / ln(7) + 1 Divide both sides by 2: x = (ln(27) / ln(7) + 1) / 2 Using a calculator,…
   Full step-by-step solution

   Step 1: Take the natural logarithm of both sides: ln(7^(2x - 1)) = ln(27) Step 2: Use the power rule of logarithms: (2x - 1) * ln(7) = ln(27) Step 3: Divide both sides by ln(7): 2x - 1 = ln(27) / ln(7) Step 4: Add 1 to both sides: 2x = ln(27) / ln(7) + 1 Step 5: Divide both sides by 2: x = (ln(27) / ln(7) + 1) / 2 Step 6: Using a calculator, ln(27) ≈ 3.2958, ln(7) ≈ 1.9459, so ln(27)/ln(7) ≈ 1.693. Then x ≈ (1.693 + 1) / 2 = 2.693 / 2 = 1.3465. Rounded to three decimal places, x ≈ 1.347.

6. log₂(2x + 6) = 5 Answer: 13 Solution: Start with the equation: log₂(2x + 6) = 5 Convert to exponential form: 2^5 = 2x + 6 Calculate 2^5: 32 = 2x + 6 Subtract 6 from both sides: 32 - 6 = 2x → 26 = 2x Divide both sides by 2: 26 ÷ 2 = x → x = 13 The answer is 13.
   Full step-by-step solution

   Step 1: Start with the equation: log₂(2x + 6) = 5 Step 2: Convert to exponential form: 2^5 = 2x + 6 Step 3: Calculate 2^5: 32 = 2x + 6 Step 4: Subtract 6 from both sides: 32 - 6 = 2x → 26 = 2x Step 5: Divide both sides by 2: 26 ÷ 2 = x → x = 13 The answer is 13.

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (0,8). A circle is inscribed in this triangle, tangent to all three sides. Visualize this configuration and determine the radius of the inscribed circle. Answer: 2 Solution: Identify the triangle's side lengths. The vertices are at (0,0), (6,0), and (0,8), so the sides are: horizontal side = 6 units, vertical side = 8 units, and hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 units.
   Full step-by-step solution

   Step 1: Identify the triangle's side lengths. The vertices are at (0,0), (6,0), and (0,8), so the sides are: horizontal side = 6 units, vertical side = 8 units, and hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 units. Step 2: Calculate the triangle's area. Area = (1/2) * base * height = (1/2) * 6 * 8 = 24 square units. Step 3: Calculate the triangle's perimeter (semi-perimeter). Perimeter = 6 + 8 + 10 = 24 units. Semi-perimeter (s) = 24/2 = 12 units. Step 4: Use the formula for the radius (r) of an inscribed circle in a triangle: r = Area / s = 24 / 12 = 2 units. The answer is 2.

8. A radioactive substance decays according to the exponential model N(t) = N₀e^(-kt), where N₀ is the initial amount. After 3 hours, only 25% of the original substance remains. A scientist observes the decay pattern on a graph showing exponential decay, with the curve passing through points representing the substance amount at various times. What is the decay constant k? Answer: ln(4)/3 Solution: N(t) = N₀ e^(-kt) After 3 hours, 25% of the original substance remains.
   Full step-by-step solution

   Let's solve step by step. We are given: N(t) = N₀ e^(-kt) After 3 hours, 25% of the original substance remains. That means: N(3) = 0.25 N₀ --- **Step 1: Substitute into the formula** N(3) = N₀ e^(-k * 3) = 0.25 N₀ --- **Step 2: Divide both sides by N₀** e^(-3k) = 0.25 --- **Step 3: Rewrite 0.25 as a fraction** 0.25 = 1/4 So: e^(-3k) = 1/4 --- **Step 4: Take natural logarithm of both sides** ln(e^(-3k)) = ln(1/4) -3k = ln(1/4) --- **Step 5: Use logarithm property** ln(1/4) = ln(1) - ln(4) = 0 - ln(4) = -ln(4) So: -3k = -ln(4) --- **Step 6: Solve for k** Multiply both sides by -1: 3k = ln(4) k = ln(4) / 3 --- **Final answer:** k = ln(4)/3