Logarithms Solve Exponential

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Logarithms Solve Exponential · Grade 11 · Worksheet 3

1. A pharmaceutical company is testing a new drug that decays exponentially in the bloodstream. The concentration C(t) in milligrams per liter is modeled by C(t) = 80 × 2^(-0.1t), where t is time in hours. The drug becomes ineffective when its concentration drops below 5 mg/L. How many hours will it take for the drug to reach this ineffective concentration level? Answer: 40 Solution: Set up the equation with the given ineffective concentration: 80 × 2^(-0.1t) = 5 Divide both sides by 80 to isolate the exponential term: 2^(-0.1t) = 5/80 Simplify the fraction: 2^(-0.1t) = 1/16 Recognize that 1/16 = 2^(-4), so: 2^(-0.1t) = 2^(-4) Since the bases are equal, set the exponents…
   Full step-by-step solution

   Step 1: Set up the equation with the given ineffective concentration: 80 × 2^(-0.1t) = 5 Step 2: Divide both sides by 80 to isolate the exponential term: 2^(-0.1t) = 5/80 Step 3: Simplify the fraction: 2^(-0.1t) = 1/16 Step 4: Recognize that 1/16 = 2^(-4), so: 2^(-0.1t) = 2^(-4) Step 5: Since the bases are equal, set the exponents equal: -0.1t = -4 Step 6: Divide both sides by -0.1: t = (-4)/(-0.1) Step 7: Calculate: t = 40 The answer is 40 hours.

2. Dr. Chen is studying bacterial growth in her lab. She observes that a colony of bacteria doubles in size every 4 hours. If she starts with 500 bacteria, how many hours will it take for the colony to reach 32,000 bacteria? Use logarithms to solve this exponential growth problem. Answer: 24 Solution: We start with 500 bacteria. The colony doubles every 4 hours. We want the time \( t \) (in hours) to reach 32,000 bacteria.
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Understand the problem** We start with 500 bacteria. The colony doubles every 4 hours. We want the time \( t \) (in hours) to reach 32,000 bacteria. --- **Step 2: Write the exponential growth formula** The general formula for exponential growth is: \[ N(t) = N_0 \times 2^{t / d} \] where: \( N_0 \) = initial number of bacteria = 500 \( d \) = doubling time = 4 hours \( N(t) \) = number of bacteria after \( t \) hours = 32,000 So: \[ 32,000 = 500 \times 2^{t / 4} \] --- **Step 3: Isolate the exponential term** Divide both sides by 500: \[ \frac{32,000}{500} = 2^{t / 4} \] \[ 64 = 2^{t / 4} \] --- **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), we can equate the exponents: \[ \frac{t}{4} = 6 \] --- **Step 6: Solve for \( t \)** Multiply both sides by 4: \[ t = 6 \times 4 = 24 \] --- **Step 7: Conclusion** It will take **24 hours** for the colony to grow from 500 to 32,000 bacteria. --- **Final answer:** 24

3. 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

4. 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.

5. 7^(x - 3) = 45 Answer: x = 3 + log_7(45) ≈ 4.956 Solution: Take the natural logarithm (or common logarithm) of both sides: ln(7^(x-3)) = ln(45). Divide both sides by ln(7): x - 3 = ln(45) / ln(7). Add 3 to both sides: x = 3 + ln(45) / ln(7).
   Full step-by-step solution

   Step 1: Take the natural logarithm (or common logarithm) of both sides: ln(7^(x-3)) = ln(45). Step 2: Apply the power rule of logarithms: (x - 3) * ln(7) = ln(45). Step 3: Divide both sides by ln(7): x - 3 = ln(45) / ln(7). Step 4: Add 3 to both sides: x = 3 + ln(45) / ln(7). Step 5: Calculate using a calculator: ln(45) ≈ 3.80666, ln(7) ≈ 1.94591, so ln(45)/ln(7) ≈ 1.956. Then x ≈ 3 + 1.956 = 4.956. The exact answer is x = 3 + log_7(45), and the approximate decimal answer is x ≈ 4.956.

6. A sound wave's intensity I is measured in decibels using the formula β = 10 log(I/I₀), where I₀ = 10⁻¹² W/m² is the reference intensity. If a rock concert measures 115 decibels, what is the intensity I of the sound wave in W/m²? Answer: 0.316227766 Solution: Write the given formula with the known values: 115 = 10 log(I/10⁻¹²) Divide both sides by 10: 11.5 = log(I/10⁻¹²) Convert from logarithmic to exponential form: I/10⁻¹² = 10¹¹·⁵ Multiply both sides by 10⁻¹²: I = 10¹¹·⁵ × 10⁻¹² Simplify using exponent rules: I = 10¹¹·⁵⁻¹² = 10⁻⁰·⁵ Calculate 10⁻⁰·⁵…
   Full step-by-step solution

   Step 1: Write the given formula with the known values: 115 = 10 log(I/10⁻¹²) Step 2: Divide both sides by 10: 11.5 = log(I/10⁻¹²) Step 3: Convert from logarithmic to exponential form: I/10⁻¹² = 10¹¹·⁵ Step 4: Multiply both sides by 10⁻¹²: I = 10¹¹·⁵ × 10⁻¹² Step 5: Simplify using exponent rules: I = 10¹¹·⁵⁻¹² = 10⁻⁰·⁵ Step 6: Calculate 10⁻⁰·⁵ = 1/10⁰·⁵ = 1/√10 ≈ 1/3.16227766 ≈ 0.316227766 The intensity is approximately 0.316 W/m².

7. 8^(x - 2) = 40 Answer: x ≈ 3.77 Solution: Take the natural log of both sides: ln(8^(x - 2)) = ln(40) Use the power rule: (x - 2) * ln(8) = ln(40) Divide both sides by ln(8): x - 2 = ln(40) / ln(8) Calculate the logs: ln(40) ≈ 3.6889, ln(8) ≈ 2.0794 Divide: 3.6889 / 2.0794 ≈ 1.774 Add 2 to both sides: x ≈ 1.774 + 2 = 3.774 The answer is…
   Full step-by-step solution

   Step 1: Take the natural log of both sides: ln(8^(x - 2)) = ln(40) Step 2: Use the power rule: (x - 2) * ln(8) = ln(40) Step 3: Divide both sides by ln(8): x - 2 = ln(40) / ln(8) Step 4: Calculate the logs: ln(40) ≈ 3.6889, ln(8) ≈ 2.0794 Step 5: Divide: 3.6889 / 2.0794 ≈ 1.774 Step 6: Add 2 to both sides: x ≈ 1.774 + 2 = 3.774 The answer is x ≈ 3.77 (rounded to two decimal places).

8. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,8). The hypotenuse has length 17 units. Using the Pythagorean theorem, determine the value of x. Answer: 15 Solution: - A = (0, 0) - B = (x, 0) - C = (0, 8) Since B is on the x-axis and C is on the y-axis, the right angle is at A = (0, 0). - Side AB is along the x-axis from (0, 0) to (x, 0). Length of AB = |x - 0| = x.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle's vertices** We have a right triangle with vertices at: - A = (0, 0) - B = (x, 0) - C = (0, 8) Since B is on the x-axis and C is on the y-axis, the right angle is at A = (0, 0). --- **Step 2: Identify the legs of the triangle** - Side AB is along the x-axis from (0, 0) to (x, 0). Length of AB = |x - 0| = x. - Side AC is along the y-axis from (0, 0) to (0, 8). Length of AC = 8. - Side BC is the hypotenuse, length = 17. --- **Step 3: Apply the Pythagorean theorem** Pythagorean theorem: (leg1)^2 + (leg2)^2 = (hypotenuse)^2 Here: AB^2 + AC^2 = BC^2 So: x^2 + 8^2 = 17^2 --- **Step 4: Calculate the squares** 8^2 = 64 17^2 = 289 So: x^2 + 64 = 289 --- **Step 5: Solve for x^2** x^2 = 289 - 64 x^2 = 225 --- **Step 6: Solve for x** x = sqrt(225) x = 15 or x = -15 But since x is a length (distance from (0,0) to (x,0) along the x-axis), we take the positive value: x = 15 --- **Final answer:** x = 15