Logarithms Solve Exponential

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Logarithms Solve Exponential · Grade 11 · Worksheet 2

1. 4^(2x - 1) = 60 Answer: x ≈ 1.979 Solution: Take the natural logarithm of both sides: ln(4^(2x - 1)) = ln(60) Apply the power rule: (2x - 1) ln(4) = ln(60) Divide both sides by ln(4): 2x - 1 = ln(60) / ln(4) Calculate the logarithms: ln(60) ≈ 4.0943, ln(4) ≈ 1.3863, so ln(60)/ln(4) ≈ 2.953 Add 1 to both sides: 2x ≈ 2.953 + 1 = 3.953…
   Full step-by-step solution

   Step 1: Take the natural logarithm of both sides: ln(4^(2x - 1)) = ln(60) Step 2: Apply the power rule: (2x - 1) ln(4) = ln(60) Step 3: Divide both sides by ln(4): 2x - 1 = ln(60) / ln(4) Step 4: Calculate the logarithms: ln(60) ≈ 4.0943, ln(4) ≈ 1.3863, so ln(60)/ln(4) ≈ 2.953 Step 5: Add 1 to both sides: 2x ≈ 2.953 + 1 = 3.953 Step 6: Divide by 2: x ≈ 3.953 / 2 = 1.9765 Step 7: Round to three decimal places: x ≈ 1.979 The answer is x ≈ 1.979.

2. 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.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (0,12). The hypotenuse extends from (5,0) to (0,12). A student draws a circle that circumscribes this triangle, passing through all three vertices. What is the area of this circumscribed circle? Answer: 42.25π Solution: For any right triangle, the hypotenuse is the diameter of the circumscribed circle. Calculate the length of the hypotenuse using the distance formula between points (5,0) and (0,12).
   Full step-by-step solution

   Step 1: For any right triangle, the hypotenuse is the diameter of the circumscribed circle. Step 2: Calculate the length of the hypotenuse using the distance formula between points (5,0) and (0,12). Step 3: Distance = sqrt((5-0)^2 + (0-12)^2) = sqrt(25 + 144) = sqrt(169) = 13 Step 4: The diameter of the circle is 13, so the radius is 13/2 = 6.5 Step 5: Area of circle = π × radius^2 = π × (6.5)^2 = π × 42.25 Step 6: The area is 42.25π

4. log₃(4x - 7) = 2 Answer: 4 Solution: Convert the logarithmic equation to exponential form. log₃(4x - 7) = 2 means 3² = 4x - 7. Calculate 3².
   Full step-by-step solution

   Step 1: Convert the logarithmic equation to exponential form. log₃(4x - 7) = 2 means 3² = 4x - 7. Step 2: Calculate 3². 3² = 9. Step 3: Substitute this value into the equation. 9 = 4x - 7. Step 4: Add 7 to both sides to isolate the term with x. 9 + 7 = 4x - 7 + 7 16 = 4x. Step 5: Divide both sides by 4 to solve for x. 16 / 4 = 4x / 4 4 = x. The solution is x = 4.

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. Isabella is monitoring the growth of a colony of bacteria in a laboratory experiment. The population of bacteria, in thousands, is modeled by the function P(t) = 12 × 7^(0.02t), where t is the number of hours since the experiment began. The laboratory's safety protocol requires the experiment to be terminated when the bacterial population reaches 84,000. How many hours will it take for the population to reach this threshold? Use logarithms to solve. Answer: 50 Solution: Set up the equation: 12 × 7^(0.02t) = 84 Divide both sides by 12: 7^(0.02t) = 84 / 12 = 7 Take the common logarithm of both sides: log(7^(0.02t)) = log(7) Apply the power rule: 0.02t × log(7) = log(7) Divide both sides by log(7): 0.02t = 1 Divide both sides by 0.02: t = 1 / 0.02 = 50 The answer…
   Full step-by-step solution

   Step 1: Set up the equation: 12 × 7^(0.02t) = 84 Step 2: Divide both sides by 12: 7^(0.02t) = 84 / 12 = 7 Step 3: Take the common logarithm of both sides: log(7^(0.02t)) = log(7) Step 4: Apply the power rule: 0.02t × log(7) = log(7) Step 5: Divide both sides by log(7): 0.02t = 1 Step 6: Divide both sides by 0.02: t = 1 / 0.02 = 50 The answer is 50 hours.

7. 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.

8. Isabella is studying the spread of a new viral infection in a controlled environment. The number of infected individuals, I(t), after t days is modeled by the function I(t) = 15 × 4^(0.1t). The health authorities will declare a health emergency when the number of infected individuals reaches 15,360. How many days will it take for the infection to reach this critical level? Use logarithms to solve. Answer: 50 Solution: Set up the equation: 15 × 4^(0.1t) = 15,360. Divide both sides by 15 to isolate the exponential: 4^(0.1t) = 15,360 / 15 = 1,024. Take the common logarithm (log) of both sides: log(4^(0.1t)) = log(1,024).
   Full step-by-step solution

   Step 1: Set up the equation: 15 × 4^(0.1t) = 15,360. Step 2: Divide both sides by 15 to isolate the exponential: 4^(0.1t) = 15,360 / 15 = 1,024. Step 3: Take the common logarithm (log) of both sides: log(4^(0.1t)) = log(1,024). Step 4: Apply the power rule of logarithms: 0.1t × log(4) = log(1,024). Step 5: Recognize that 1,024 = 4^5, so log(1,024) = log(4^5) = 5 × log(4). Step 6: Substitute: 0.1t × log(4) = 5 × log(4). Step 7: Divide both sides by log(4) (which is positive): 0.1t = 5. Step 8: Divide both sides by 0.1: t = 5 / 0.1 = 50. The answer is 50 days.