Logarithmic Form Solutions

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Logarithmic Form Solutions · Grade 11 · Worksheet 2

1. Lena is studying sound intensity levels in her physics class. She learns that the decibel level of a sound is given by the formula dB = 10 × log(I/I₀), where I is the sound intensity and I₀ is the reference intensity of 10⁻¹² W/m². If Lena measures a sound at 85 decibels, what is the ratio of its intensity to the reference intensity? Express your answer in logarithmic form. Answer: log(I/I₀) = 8.5 Solution: Start with the given formula: dB = 10 × log(I/I₀) Substitute the known decibel value: 85 = 10 × log(I/I₀) Divide both sides by 10 to isolate the logarithm: 85/10 = log(I/I₀) Simplify the division: 8.5 = log(I/I₀) The ratio in logarithmic form is: log(I/I₀) = 8.5
   Full step-by-step solution

   Step 1: Start with the given formula: dB = 10 × log(I/I₀) Step 2: Substitute the known decibel value: 85 = 10 × log(I/I₀) Step 3: Divide both sides by 10 to isolate the logarithm: 85/10 = log(I/I₀) Step 4: Simplify the division: 8.5 = log(I/I₀) Step 5: The ratio in logarithmic form is: log(I/I₀) = 8.5

2. 7^(x+2) = 49. Express solution using logarithms Answer: x = log_7(49) - 2 Solution: Start with the equation 7^(x+2) = 49 Take the logarithm base 7 of both sides: log_7(7^(x+2)) = log_7(49) Simplify the left side using the property log_b(b^a) = a: x+2 = log_7(49) Subtract 2 from both sides: x = log_7(49) - 2 The solution in logarithmic form is x = log_7(49) - 2
   Full step-by-step solution

   Step 1: Start with the equation 7^(x+2) = 49 Step 2: Take the logarithm base 7 of both sides: log_7(7^(x+2)) = log_7(49) Step 3: Simplify the left side using the property log_b(b^a) = a: x+2 = log_7(49) Step 4: Subtract 2 from both sides: x = log_7(49) - 2 Step 5: The solution in logarithmic form is x = log_7(49) - 2

3. 5^(3x-1) = 125. Express solution using logarithms. Answer: x = (log₅(125) + 1)/3 Solution: Start with the equation: 5^(3x-1) = 125 Convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c). Here a = 5, b = 3x-1, c = 125.
   Full step-by-step solution

   Step 1: Start with the equation: 5^(3x-1) = 125 Step 2: Convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c). Here a = 5, b = 3x-1, c = 125. Step 3: This gives: 3x - 1 = log₅(125) Step 4: Add 1 to both sides: 3x = log₅(125) + 1 Step 5: Divide both sides by 3: x = (log₅(125) + 1)/3 The solution expressed in logarithmic form is x = (log₅(125) + 1)/3.

4. log₄(64) = ? Answer: 3 Solution: We need to find the exponent x such that 4^x = 64 Recognize that 64 can be written as 4^3 because 4 × 4 × 4 = 16 × 4 = 64 Therefore, 4^3 = 64 This means log₄(64) = 3 The answer is 3.
   Full step-by-step solution

   Step 1: We need to find the exponent x such that 4^x = 64 Step 2: Recognize that 64 can be written as 4^3 because 4 × 4 × 4 = 16 × 4 = 64 Step 3: Therefore, 4^3 = 64 Step 4: This means log₄(64) = 3 The answer is 3.

5. A biologist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and it grows at a continuous rate of 8% per hour. The biologist needs to determine how many hours it will take for the population to reach 2,000 bacteria. Write your answer in logarithmic form. Answer: t = ln(4) / 0.08 Solution: We are told the initial population is 500 bacteria and it grows at a continuous rate of 8% per hour. We want the time t for the population to reach 2000 bacteria.
   Full step-by-step solution

   We are told the initial population is 500 bacteria and it grows at a continuous rate of 8% per hour. We want the time t for the population to reach 2000 bacteria. --- **Step 1: Set up the continuous growth formula** For continuous growth at rate r (as a decimal), the population at time t is: P(t) = P0 * e^(r t) Here: P0 = 500 r = 0.08 P(t) = 2000 So: 2000 = 500 * e^(0.08 t) --- **Step 2: Divide both sides by 500** 2000 / 500 = 4 So: 4 = e^(0.08 t) --- **Step 3: Take the natural logarithm of both sides** ln(4) = ln(e^(0.08 t)) Since ln(e^x) = x: ln(4) = 0.08 t --- **Step 4: Solve for t** t = ln(4) / 0.08 --- **Final answer:** t = ln(4) / 0.08

6. 4^(2x) = 48. Express solution using logarithms. Answer: x = log₄(48)/2 Solution: Start with the equation: 4^(2x) = 48 Convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c). Here a = 4, b = 2x, c = 48.
   Full step-by-step solution

   Step 1: Start with the equation: 4^(2x) = 48 Step 2: Convert to logarithmic form using the definition: if a^b = c, then b = logₐ(c). Here a = 4, b = 2x, c = 48. Step 3: This gives: 2x = log₄(48) Step 4: Divide both sides by 2 to isolate x: x = log₄(48)/2 The solution expressed in logarithmic form is x = log₄(48)/2.

7. A financial analyst is modeling compound interest for an investment account. The account balance A(t) after t years is given by A(t) = 2500 × (1.045)^t. The analyst needs to determine how many years it will take for the investment to triple in value. Express your answer in logarithmic form. Answer: log(3)/log(1.045) Solution: Set up the equation for tripling: 2500 × (1.045)^t = 3 × 2500 Simplify by dividing both sides by 2500: (1.045)^t = 3 Take the logarithm of both sides: log((1.045)^t) = log(3) Apply the power rule of logarithms: t × log(1.045) = log(3) Solve for t: t = log(3)/log(1.045) The answer in logarithmic…
   Full step-by-step solution

   Step 1: Set up the equation for tripling: 2500 × (1.045)^t = 3 × 2500 Step 2: Simplify by dividing both sides by 2500: (1.045)^t = 3 Step 3: Take the logarithm of both sides: log((1.045)^t) = log(3) Step 4: Apply the power rule of logarithms: t × log(1.045) = log(3) Step 5: Solve for t: t = log(3)/log(1.045) The answer in logarithmic form is log(3)/log(1.045).

8. 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)) = 3 Step 2: Convert to exponential form: x(x+6) = 3³ = 27 Step 3: Expand and rearrange: x² + 6x - 27 = 0 Step 4: Factor the quadratic: (x+9)(x-3) = 0 Step 5: Solve for x: x = -9 or x = 3 Step 6: Check domain…
   Full step-by-step solution

   Step 1: Apply the product rule for logarithms: log₃(x) + log₃(x+6) = log₃(x(x+6)) = 3 Step 2: Convert to exponential form: x(x+6) = 3³ = 27 Step 3: Expand and rearrange: x² + 6x - 27 = 0 Step 4: Factor the quadratic: (x+9)(x-3) = 0 Step 5: Solve for x: x = -9 or x = 3 Step 6: Check domain restrictions: log₃(x) requires x > 0, so x = -9 is extraneous The valid solution is x = 3.