Logarithm Properties

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Logarithm Properties · Grade 11 · Worksheet 3

1. log₇(343x) + log₇(x/7) = 5 Answer: x = 7 Solution: Simplify inside the logarithm: 343/7 = 49, so we have log₇(49x²). The equation becomes log₇(49x²) = 5. Rewrite in exponential form: 7⁵ = 49x².
   Full step-by-step solution

   Step 1: Apply the product property: log₇(343x) + log₇(x/7) = log₇(343x * x/7) = log₇(343x²/7). Step 2: Simplify inside the logarithm: 343/7 = 49, so we have log₇(49x²). Step 3: The equation becomes log₇(49x²) = 5. Step 4: Rewrite in exponential form: 7⁵ = 49x². Step 5: Compute 7⁵ = 16807, so 16807 = 49x². Step 6: Divide both sides by 49: x² = 16807 / 49 = 343. Step 7: Take the square root: x = √343 = 7 (since x > 0 for the logarithm to be defined). The answer is x = 7.

2. Dr. Rodriguez is analyzing sound intensity levels in a concert hall. The decibel level of a sound is given by the formula L = 10 log(I/I₀), where I is the sound intensity and I₀ is the reference intensity of 10⁻¹² W/m². During a rock concert, the sound intensity reaches 0.01 W/m². What is the sound intensity level in decibels? Answer: 100 Solution: Write the decibel formula: L = 10 log(I/I₀) Substitute the given values: I = 0.01 W/m² and I₀ = 10⁻¹² W/m² Calculate the intensity ratio: I/I₀ = 0.01 / 10⁻¹² = 10⁻² / 10⁻¹² = 10¹⁰ Substitute into the formula: L = 10 log(10¹⁰) Use the logarithm property: log(10¹⁰) = 10 Calculate: L = 10 × 10 =…
   Full step-by-step solution

   Step 1: Write the decibel formula: L = 10 log(I/I₀) Step 2: Substitute the given values: I = 0.01 W/m² and I₀ = 10⁻¹² W/m² Step 3: Calculate the intensity ratio: I/I₀ = 0.01 / 10⁻¹² = 10⁻² / 10⁻¹² = 10¹⁰ Step 4: Substitute into the formula: L = 10 log(10¹⁰) Step 5: Use the logarithm property: log(10¹⁰) = 10 Step 6: Calculate: L = 10 × 10 = 100 The sound intensity level is 100 decibels.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (8,6). A circle is inscribed within this triangle, tangent to all three sides. What is the exact area of the circle? (Express your answer in terms of π) Answer: 4π Solution: Identify the triangle's dimensions. The vertices are at (0,0), (8,0), and (8,6), so the legs are 8 units and 6 units.
   Full step-by-step solution

   Step 1: Identify the triangle's dimensions. The vertices are at (0,0), (8,0), and (8,6), so the legs are 8 units and 6 units. Step 2: Calculate the hypotenuse using the Pythagorean theorem: sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 units. Step 3: For a right triangle, the inradius (r) can be found using the formula: r = (a + b - c)/2, where a and b are the legs and c is the hypotenuse. Step 4: Substitute the values: r = (8 + 6 - 10)/2 = (4)/2 = 2 units. Step 5: Calculate the area of the circle using A = πr^2: A = π(2)^2 = 4π. The exact area of the inscribed circle is 4π.

4. log₇(343) + log₉(81) - log₅(125) = ? Answer: 2 Solution: Evaluate log₇(343) 7^x = 343 7^3 = 343 So log₇(343) = 3 Evaluate log₉(81) 9^x = 81 9^2 = 81 So log₉(81) = 2 Evaluate log₅(125) 5^x = 125 5^3 = 125 So log₅(125) = 3 Substitute the values into the original expression 3 + 2 - 3 = 5 - 3 = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Evaluate log₇(343) 7^x = 343 7^3 = 343 So log₇(343) = 3 Step 2: Evaluate log₉(81) 9^x = 81 9^2 = 81 So log₉(81) = 2 Step 3: Evaluate log₅(125) 5^x = 125 5^3 = 125 So log₅(125) = 3 Step 4: Substitute the values into the original expression 3 + 2 - 3 = 5 - 3 = 2 The answer is 2.

5. log₃(81) - log₂(32) + log₅(125) = ? Answer: 2 Solution: Evaluate log₃(81) Since 3⁴ = 81, log₃(81) = 4 Evaluate log₂(32) Since 2⁵ = 32, log₂(32) = 5 Evaluate log₅(125) Since 5³ = 125, log₅(125) = 3 Substitute the values into the original expression 4 - 5 + 3 4 - 5 = -1 -1 + 3 = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Evaluate log₃(81) Since 3⁴ = 81, log₃(81) = 4 Step 2: Evaluate log₂(32) Since 2⁵ = 32, log₂(32) = 5 Step 3: Evaluate log₅(125) Since 5³ = 125, log₅(125) = 3 Step 4: Substitute the values into the original expression 4 - 5 + 3 Step 5: Perform the operations from left to right 4 - 5 = -1 -1 + 3 = 2 The answer is 2.

6. Charlotte is a chemist analyzing the pH of two different soil samples. The pH of a solution is defined as pH = -log[H⁺], where [H⁺] is the hydrogen ion concentration in moles per liter. Sample A has a hydrogen ion concentration of 3.2 × 10⁻⁷ M, and Sample B has a hydrogen ion concentration of 2.0 × 10⁻⁸ M. Using properties of logarithms, determine how many times more acidic Sample A is compared to Sample B. Answer: 16 Solution: Write the ratio of hydrogen ion concentrations: ratio = [H⁺]_A / [H⁺]_B = (3.2 × 10⁻⁷) / (2.0 × 10⁻⁸). Divide the coefficients: 3.2 / 2.0 = 1.6. Divide the powers of 10: 10⁻⁷ / 10⁻⁸ = 10^(-7 - (-8)) = 10^1 = 10.
   Full step-by-step solution

   Step 1: Write the ratio of hydrogen ion concentrations: ratio = [H⁺]_A / [H⁺]_B = (3.2 × 10⁻⁷) / (2.0 × 10⁻⁸). Step 2: Divide the coefficients: 3.2 / 2.0 = 1.6. Step 3: Divide the powers of 10: 10⁻⁷ / 10⁻⁸ = 10^(-7 - (-8)) = 10^1 = 10. Step 4: Multiply the results: ratio = 1.6 × 10 = 16. Step 5: Therefore, Sample A is 16 times more acidic than Sample B. The answer is 16.

7. log₃(81x) + log₃(x/9) = 5 Answer: x = 9√3 Solution: Simplify inside the logarithm: (81x)(x/9) = 81x²/9 = 9x². Rewrite in exponential form: 3⁵ = 9x². Calculate 3⁵ = 243, so 243 = 9x².
   Full step-by-step solution

   Step 1: Apply the product rule: log₃(81x) + log₃(x/9) = log₃((81x)(x/9)). Step 2: Simplify inside the logarithm: (81x)(x/9) = 81x²/9 = 9x². So the equation becomes log₃(9x²) = 5. Step 3: Rewrite in exponential form: 3⁵ = 9x². Step 4: Calculate 3⁵ = 243, so 243 = 9x². Step 5: Divide both sides by 9: x² = 27. Step 6: Take the square root: x = √27 = 3√3 (since x > 0 for the original logarithms to be defined). The answer is x = 3√3.

8. Given that log₂(3) ≈ 1.585 and log₂(5) ≈ 2.322, use properties of logarithms to evaluate log₂(135) to the nearest hundredth. Answer: 7.08 Solution: Factor 135 to find numbers related to the given logarithms.
   Full step-by-step solution

   Step 1: Factor 135 to find numbers related to the given logarithms. 135 = 27 × 5 = 3³ × 5 Step 2: Apply the logarithm property for products: log₂(135) = log₂(3³ × 5) = log₂(3³) + log₂(5) Step 3: Apply the power rule: log₂(3³) = 3 × log₂(3) = 3 × 1.585 = 4.755 Step 4: Add the results: 4.755 + 2.322 = 7.077 Step 5: Round to the nearest hundredth: 7.08 The answer is 7.08.