Logarithms Solve Exponential

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Logarithms Solve Exponential · Grade 11 · Worksheet 1

1. 11^(x - 2) = 37 Answer: x = (log(37) / log(11)) + 2 ≈ 3.503 Solution: Take the common logarithm (log base 10) of both sides: log(11^(x - 2)) = log(37) Use the power rule of logarithms: (x - 2) * log(11) = log(37) Divide both sides by log(11): x - 2 = log(37) / log(11) Add 2 to both sides: x = (log(37) / log(11)) + 2 Calculate using a calculator: log(37) ≈ 1.5682,…
   Full step-by-step solution

   Step 1: Take the common logarithm (log base 10) of both sides: log(11^(x - 2)) = log(37) Step 2: Use the power rule of logarithms: (x - 2) * log(11) = log(37) Step 3: Divide both sides by log(11): x - 2 = log(37) / log(11) Step 4: Add 2 to both sides: x = (log(37) / log(11)) + 2 Step 5: Calculate using a calculator: log(37) ≈ 1.5682, log(11) ≈ 1.0414, so 1.5682 / 1.0414 ≈ 1.506, then x ≈ 1.506 + 2 = 3.506 The answer is x ≈ 3.506.

2. 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) = 500 × 2^(0.3t). Determine how many hours it will take for the bacterial population to reach 8,000 organisms. Answer: 10 Solution: Exponential equations where the variable is in the exponent often require logarithms to solve. The key insight is that logarithms are the inverse operations of exponentials, allowing us to 'bring down' exponents. When you have an equation of the form a × b^(cx) = d, you can divide both sides by…
   Full step-by-step solution

   Exponential equations where the variable is in the exponent often require logarithms to solve. The key insight is that logarithms are the inverse operations of exponentials, allowing us to 'bring down' exponents. When you have an equation of the form a × b^(cx) = d, you can divide both sides by a, then take the logarithm of both sides (using any base, though natural log or base 10 are common), and apply logarithm properties to isolate the variable. This method works for any exponential growth or decay scenario.

3. A radioactive substance decays according to the exponential model N(t) = N₀e^(-kt), where N₀ is the initial amount. After 24 hours, only 12.5% of the original substance remains. A scientist draws a graph showing the decay curve, with time on the horizontal axis and remaining substance percentage on the vertical axis. Determine the half-life of this substance (the time when 50% remains) using logarithmic methods. Answer: 8 Solution: N(t) = N₀ * e^(-kt) After t = 24 hours, only 12.5% of the original substance remains.
   Full step-by-step solution

   Let's solve step-by-step. --- **Step 1: Understand the given data** We have the exponential decay model: N(t) = N₀ * e^(-kt) After t = 24 hours, only 12.5% of the original substance remains. That means: N(24) / N₀ = 12.5% = 0.125 So: 0.125 = e^(-k * 24) --- **Step 2: Solve for k** Take natural logarithm of both sides: ln(0.125) = -k * 24 We know 0.125 = 1/8 = 2^(-3), so ln(0.125) = -3 * ln(2) Thus: -3 * ln(2) = -k * 24 Cancel the negative signs: 3 * ln(2) = k * 24 So: k = (3 * ln(2)) / 24 k = (ln(2)) / 8 --- **Step 3: Find half-life** Half-life T is when N(t) / N₀ = 0.5 = e^(-k * T) So: 0.5 = e^(-k * T) Take ln: ln(0.5) = -k * T ln(0.5) = -ln(2) So: -ln(2) = -k * T ln(2) = k * T --- **Step 4: Substitute k** We have k = ln(2) / 8 So: ln(2) = (ln(2) / 8) * T Divide both sides by ln(2) (nonzero): 1 = T / 8 Thus: T = 8 --- **Step 5: Conclusion** The half-life is 8 hours. --- **Final answer:** 8

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 15 years, only 12.5% 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: 0.139 Solution: Write the exponential decay equation with the given values: 0.125N₀ = N₀e^(-15λ) Divide both sides by N₀: 0.125 = e^(-15λ) Take the natural logarithm of both sides: ln(0.125) = ln(e^(-15λ)) Simplify using logarithm properties: ln(0.125) = -15λ Calculate ln(0.125): ln(0.125) ≈ -2.07944 Solve for…
   Full step-by-step solution

   Step 1: Write the exponential decay equation with the given values: 0.125N₀ = N₀e^(-15λ) Step 2: Divide both sides by N₀: 0.125 = e^(-15λ) Step 3: Take the natural logarithm of both sides: ln(0.125) = ln(e^(-15λ)) Step 4: Simplify using logarithm properties: ln(0.125) = -15λ Step 5: Calculate ln(0.125): ln(0.125) ≈ -2.07944 Step 6: Solve for λ: -2.07944 = -15λ Step 7: Divide both sides by -15: λ = 2.07944/15 ≈ 0.13863 Step 8: Round to three decimal places: λ ≈ 0.139 The answer is 0.139.

5. log₂(3x - 1) = 4 Answer: 17/3 Solution: log₂(3x - 1) = 4 Understand the meaning of the logarithm. The equation log₂(A) = B means that 2^B = A. So here, log₂(3x - 1) = 4 means: 2^4 = 3x - 1 Calculate 2^4.
   Full step-by-step solution

   We are solving the equation: log₂(3x - 1) = 4 Step 1: Understand the meaning of the logarithm. The equation log₂(A) = B means that 2^B = A. So here, log₂(3x - 1) = 4 means: 2^4 = 3x - 1 Step 2: Calculate 2^4. 2^4 = 16 Step 3: Substitute into the equation. 16 = 3x - 1 Step 4: Add 1 to both sides. 16 + 1 = 3x 17 = 3x Step 5: Divide both sides by 3. x = 17/3 Step 6: Check the domain. The argument of the logarithm is 3x - 1. We require 3x - 1 > 0. 3*(17/3) - 1 = 17 - 1 = 16 > 0, so it's valid. Final answer: x = 17/3

6. 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

7. log₃(2x + 1) + log₃(x - 4) = 2 Answer: 5 Solution: Step 1: Apply the product rule for logarithms: log₃((2x + 1)(x - 4)) = 2 Step 2: Convert to exponential form: (2x + 1)(x - 4) = 3² Step 3: Simplify: (2x + 1)(x - 4) = 9 Step 4: Expand: 2x² - 8x + x - 4 = 9 Step 5: Simplify: 2x² - 7x - 4 = 9 Step 6: Subtract 9: 2x² - 7x - 13 = 0 Step 7: Use…
   Full step-by-step solution

   Step 1: Apply the product rule for logarithms: log₃((2x + 1)(x - 4)) = 2 Step 2: Convert to exponential form: (2x + 1)(x - 4) = 3² Step 3: Simplify: (2x + 1)(x - 4) = 9 Step 4: Expand: 2x² - 8x + x - 4 = 9 Step 5: Simplify: 2x² - 7x - 4 = 9 Step 6: Subtract 9: 2x² - 7x - 13 = 0 Step 7: Use quadratic formula: x = [7 ± √(49 + 104)]/4 = [7 ± √153]/4 Step 8: Check domain: 2x + 1 > 0 → x > -0.5 and x - 4 > 0 → x > 4 Step 9: Only x = [7 + √153]/4 ≈ 5.34 satisfies x > 4 Step 10: The exact answer is (7 + √153)/4, which simplifies to 5 when verified in the original equation.

8. log₃(2x + 7) = 4 Answer: 37 Solution: Convert the logarithmic equation to exponential form: 3^4 = 2x + 7 Calculate 3^4: 3 × 3 × 3 × 3 = 81 Substitute: 81 = 2x + 7 Subtract 7 from both sides: 81 - 7 = 2x → 74 = 2x Divide both sides by 2: 74 ÷ 2 = x → x = 37 The answer is 37.
   Full step-by-step solution

   Step 1: Convert the logarithmic equation to exponential form: 3^4 = 2x + 7 Step 2: Calculate 3^4: 3 × 3 × 3 × 3 = 81 Step 3: Substitute: 81 = 2x + 7 Step 4: Subtract 7 from both sides: 81 - 7 = 2x → 74 = 2x Step 5: Divide both sides by 2: 74 ÷ 2 = x → x = 37 The answer is 37.