Question

on an island, the deer population has been decreasing by 4% per year (t). the current population is 792 deer. assume that the deer population continues to decrease by 4% per year. which of the following logarithmic equations best models the number of years t it will take for the deer population to decrease from 792 deer to 674 deer? a. $t = \log_{1.04} 0.85$ b. $t = \log_{0.96} 0.04$ c. $t = \log_{0.96} 1.04$ d. $t = \log_{0.96} 0.85$

Explanation:

Step1: Recall Exponential Decay Formula

The formula for exponential decay is \( P(t) = P_0(1 - r)^t \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the rate of decay, and \( t \) is time in years. Here, \( P_0 = 792 \), \( r = 0.04 \) (4% decay), so \( 1 - r = 0.96 \). Wait, the final population is 674? Wait, no, maybe I miscalculated. Wait, \( P(t) = 674 \), \( P_0 = 792 \), so \( 674 = 792(0.96)^t \). Divide both sides by 792: \( \frac{674}{792} \approx 0.851 \approx 0.85 \). So \( 0.85 = (0.96)^t \). To solve for \( t \), take log base 0.96 of both sides: \( t = \log_{0.96} 0.85 \). Wait, but the options: Let's check the options again. Wait, maybe I made a mistake. Wait, the options are A: \( t = \log_{1.04} 0.85 \), B: \( t = \log_{0.96}(-0.04) \) (invalid, log of negative), C: \( t = \log_{0.96} 1.04 \), D: \( t = \log_{0.96} 0.85 \). Wait, no, wait \( 1 - r = 0.96 \), so the base is 0.96. And \( \frac{674}{792} \approx 0.85 \). So the equation is \( 0.85 = 0.96^t \), so \( t = \log_{0.96} 0.85 \), which is option D? Wait, but let's re-express. Alternatively, using change of base, but the key is the base is \( 1 - 0.04 = 0.96 \), and the argument is \( \frac{674}{792} \approx 0.85 \). So the correct logarithmic equation is \( t = \log_{0.96} 0.85 \), which is option D? Wait, but let's check the options again. Wait, the options: A is base 1.04, which is growth, so no. B has negative argument, invalid. C: log of 1.04 with base 0.96, which doesn't make sense. D: log of 0.85 with base 0.96, which matches our derivation. Wait, but maybe I miscalculated \( \frac{674}{792} \). Let's compute 674 divided by 792: 674 ÷ 792 ≈ 0.851, which is approximately 0.85. So yes, \( t = \log_{0.96} 0.85 \), which is option D? Wait, but the original problem's options: Let me check again. The user's options: A: \( t = \log_{1.04} 0.85 \), B: \( t = \log_{0.96}(-0.04) \), C: \( t = \log_{0.96} 1.04 \), D: \( t = \log_{0.96} 0.85 \). So the correct one is D, because the base is the decay factor (0.96), and the argument is the ratio of final to initial population (≈0.85), and solving \( 0.85 = 0.96^t \) gives \( t = \log_{0.96} 0.85 \).

Step2: Eliminate Invalid Options

• Option B: Logarithm of a negative number is undefined, so B is invalid.
• Option A: Base 1.04 is a growth factor (1 + 0.04), but we have decay (1 - 0.04 = 0.96), so A is incorrect.
• Option C: The argument 1.04 and base 0.96 don't relate to the population ratio, so C is incorrect.
• Option D: Base 0.96 (decay factor) and argument 0.85 (population ratio) matches the exponential decay equation solved for \( t \).

Answer:

D. \( t = \log_{0.96} 0.85 \)