Logarithmic Form Solutions

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Logarithmic Form Solutions · Grade 11 · Worksheet 3

1. 9^x = 47. Express solution using logarithms Answer: x = log₉(47) Solution: Start with the exponential equation: 9^x = 47 To express this in logarithmic form, recall that if a^b = c, then b = log_a(c) Here, a = 9, b = x, and c = 47 Therefore, x = log₉(47) The solution in logarithmic form is x = log₉(47)
   Full step-by-step solution

   Step 1: Start with the exponential equation: 9^x = 47 Step 2: To express this in logarithmic form, recall that if a^b = c, then b = log_a(c) Step 3: Here, a = 9, b = x, and c = 47 Step 4: Therefore, x = log₉(47) The solution in logarithmic form is x = log₉(47)

2. The sound intensity level in decibels 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. If a rock concert measures 115 dB, how many times more intense is this sound than the threshold of hearing (0 dB)? Answer: 31622776 Solution: For the rock concert: 115 = 10 log(I₁/I₀) For the threshold of hearing: 0 = 10 log(I₂/I₀) From step 2: 0 = 10 log(I₂/I₀), so log(I₂/I₀) = 0, which means I₂/I₀ = 10⁰ = 1, so I₂ = I₀ From step 1: 115 = 10 log(I₁/I₀), so 11.5 = log(I₁/I₀) Convert from logarithmic form: I₁/I₀ = 10¹¹·⁵ Since I₂ = I₀,…
   Full step-by-step solution

   Step 1: For the rock concert: 115 = 10 log(I₁/I₀) Step 2: For the threshold of hearing: 0 = 10 log(I₂/I₀) Step 3: From step 2: 0 = 10 log(I₂/I₀), so log(I₂/I₀) = 0, which means I₂/I₀ = 10⁰ = 1, so I₂ = I₀ Step 4: From step 1: 115 = 10 log(I₁/I₀), so 11.5 = log(I₁/I₀) Step 5: Convert from logarithmic form: I₁/I₀ = 10¹¹·⁵ Step 6: Since I₂ = I₀, the ratio I₁/I₂ = I₁/I₀ = 10¹¹·⁵ Step 7: Calculate 10¹¹·⁵ = 10¹¹ × 10⁰·⁵ = 100,000,000,000 × 3.16227766 ≈ 316,227,766 Step 8: The rock concert is approximately 316,227,766 times more intense than the threshold of hearing. The answer is 31622776.

3. An archaeologist is analyzing the decay of carbon-14 in an ancient artifact. The artifact originally contained 120 micrograms of carbon-14, but now contains only 15 micrograms. Given that carbon-14 has a half-life of 5,730 years, determine how many years have passed since the artifact was created. Express your answer in logarithmic form. Answer: t = 5730 * ln(8)/ln(2) Solution: Use the exponential decay formula: A = A₀ * (1/2)^(t/h), where A is current amount, A₀ is original amount, t is time, h is half-life.
   Full step-by-step solution

   Step 1: Use the exponential decay formula: A = A₀ * (1/2)^(t/h), where A is current amount, A₀ is original amount, t is time, h is half-life. Step 2: Substitute known values: 15 = 120 * (1/2)^(t/5730) Step 3: Divide both sides by 120: 15/120 = (1/2)^(t/5730) Step 4: Simplify: 1/8 = (1/2)^(t/5730) Step 5: Rewrite 1/8 as (1/2)^3: (1/2)^3 = (1/2)^(t/5730) Step 6: Since the bases are equal, set exponents equal: 3 = t/5730 Step 7: Multiply both sides by 5730: t = 3 * 5730 Step 8: Express in logarithmic form using the relationship from step 5: t = 5730 * log₁/₂(1/8) Step 9: Convert to natural logarithms: t = 5730 * ln(1/8)/ln(1/2) Step 10: Simplify using logarithm properties: t = 5730 * [-ln(8)]/[-ln(2)] = 5730 * ln(8)/ln(2) The answer is t = 5730 * ln(8)/ln(2)

4. Liam is an archaeologist studying radioactive carbon-14 decay in ancient artifacts. He discovers a wooden tool that contains only 12.5% of its original carbon-14 content. Given that carbon-14 has a half-life of 5,730 years, determine how many years ago the tree used to make this tool was cut down. Express your answer in logarithmic form. Answer: t = (5730 × ln(8))/ln(2) Solution: Radioactive decay follows an exponential pattern where the remaining amount equals the initial amount multiplied by (1/2) raised to the power of (time/half-life).
   Full step-by-step solution

   Radioactive decay follows an exponential pattern where the remaining amount equals the initial amount multiplied by (1/2) raised to the power of (time/half-life). To solve for time when given a percentage remaining, we convert the percentage to a fraction, set up the exponential equation, and then use logarithms to isolate the time variable. This approach works for any radioactive substance when you know its half-life and the percentage remaining.

5. A research scientist named Dr. Chen is studying bacterial growth in a lab. She observes that a bacterial culture starts with 500 cells and grows exponentially at a rate of 8% per hour. Using the exponential growth formula N(t) = N₀e^(kt), where N₀ is the initial population and k is the growth constant, determine how many hours it will take for the bacterial population to reach 2,000 cells. Express your answer in logarithmic form. Answer: t = (ln(4)) / 0.08 Solution: Initial population: \( N_0 = 500 \) Final population: \( N(t) = 2000 \) Growth rate: 8% per hour, so \( k = 0.08 \) Formula: \( N(t) = N_0 e^{kt} \) 2000 = 500 e^{0.08 t} Divide both sides by 500 \frac{2000}{500} = e^{0.08 t} 4 = e^{0.08 t} \ln(4) = \ln(e^{0.08 t}) \ln(4) = 0.08 t because \(…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the given values** Initial population: \( N_0 = 500 \) Final population: \( N(t) = 2000 \) Growth rate: 8% per hour, so \( k = 0.08 \) Formula: \( N(t) = N_0 e^{kt} \) --- **Step 2: Substitute into the formula** \[ 2000 = 500 e^{0.08 t} \] --- **Step 3: Divide both sides by 500** \[ \frac{2000}{500} = e^{0.08 t} \] \[ 4 = e^{0.08 t} \] --- **Step 4: Take the natural logarithm of both sides** \[ \ln(4) = \ln(e^{0.08 t}) \] \[ \ln(4) = 0.08 t \] because \( \ln(e^x) = x \). --- **Step 5: Solve for t** \[ t = \frac{\ln(4)}{0.08} \] --- **Final answer:** t = (ln(4)) / 0.08

6. 8^x = 64. Express solution using logarithms Answer: x = log_8(64) Solution: Start with the exponential equation: 8^x = 64 To express this in logarithmic form, we write: x = log_8(64) This means x is the exponent we need to raise 8 to in order to get 64 The solution in logarithmic form is x = log_8(64)
   Full step-by-step solution

   Step 1: Start with the exponential equation: 8^x = 64 Step 2: To express this in logarithmic form, we write: x = log_8(64) Step 3: This means x is the exponent we need to raise 8 to in order to get 64 Step 4: The solution in logarithmic form is x = log_8(64)

7. Kaia is a marine biologist studying the growth of a coral reef. The surface area A(t) of the reef in square meters after t years is modeled by A(t) = 7 × (1.03)^t. Kaia wants to know how many years it will take for the reef to grow to a surface area of 21 square meters. Express your answer in logarithmic form. Answer: log(3) / log(1.03) Solution: Set up the equation for the final surface area: 7 × (1.03)^t = 21. Divide both sides by 7: (1.03)^t = 3. Take the logarithm of both sides (common log or natural log): log((1.03)^t) = log(3).
   Full step-by-step solution

   Step 1: Set up the equation for the final surface area: 7 × (1.03)^t = 21. Step 2: Divide both sides by 7: (1.03)^t = 3. Step 3: Take the logarithm of both sides (common log or natural log): log((1.03)^t) = log(3). Step 4: Apply the power rule of logarithms: t × log(1.03) = log(3). Step 5: Solve for t: t = log(3) / log(1.03). The answer in logarithmic form is log(3) / log(1.03).