This question tests AP Precalculus skills: using logarithms to solve half-life problems in radioactive decay. The decay formula A(t) = A₀(1/2)^(t/8) shows the amount halves every 8 days. To find when A(t) = 0.20A₀, we solve 0.20A₀ = A₀(1/2)^(t/8). Choice A is correct because dividing by A₀ gives 0.20 = (1/2)^(t/8), taking logarithms yields log(0.20) = (t/8)·log(1/2), and solving for t gives t = 8·log(0.20)/log(1/2). Choice D incorrectly uses log(2) in the denominator instead of log(1/2), which would give the negative of the correct answer. To help students: Remember that log(1/2) = -log(2), so both forms can work if signs are handled correctly. Practice recognizing the half-life formula structure with the time constant in the exponent.

This question tests AP Precalculus skills: applying logarithms to solve exponential growth equations in demographic modeling. The population model P(t) = $150,000(1.03)^t$ represents 3% annual growth from 150,000. To find when P(t) = 200,000, we solve 200,000 = $150,000(1.03)^t$. Choice A is correct because dividing by 150,000 gives 200,000/150,000 = $(1.03)^t$, then taking logarithms yields log(200,000/150,000) = t·log(1.03), so t = log(200,000/150,000)/log(1.03). Choice C incorrectly uses log(0.03) instead of log(1.03), confusing the growth rate 0.03 with the growth factor 1.03. To help students: Emphasize that for r% growth, we use the base (1 + r/100). Practice setting up ratios of final value to initial value before applying logarithms.

This question tests AP Precalculus skills: understanding and applying logarithmic functions to solve exponential growth problems. The population model P(t) = $120,000(1.03)^t$ represents exponential growth with a 3% annual increase. To find when the population doubles to 240,000, we set up 240,000 = $120,000(1.03)^t$ and solve for t. Dividing both sides by 120,000 gives 2 = $(1.03)^t$, then taking logarithms of both sides yields log(2) = t·log(1.03), so t = log(2)/log(1.03). Choice A is correct because it properly applies the power rule of logarithms and isolates t. Choice B incorrectly inverts the fraction, while Choice C attempts to use the ratio of populations directly without proper logarithmic manipulation. To help students: Emphasize the power rule $log(a^x$) = x·log(a) and practice solving exponential equations step by step.

This question tests AP Precalculus skills: solving exponential inequalities for investment growth using logarithms. The problem requires finding when an investment doubles, which is a common financial application. We need to solve $3000(1.06)^t$ ≥ 6000, which simplifies to $(1.06)^t$ ≥ 2. Choice A is correct because taking logarithms gives t·log(1.06) ≥ log(2), and since log(1.06) > 0, dividing preserves the inequality: t ≥ log(2)/log(1.06). Choice B has the wrong inequality direction, while Choice C inverts the fraction incorrectly. To help students: Emphasize that when the base is greater than 1, logarithms preserve inequality direction. Practice identifying doubling scenarios in exponential growth problems.

This question tests AP Precalculus skills: understanding and applying logarithmic functions to radioactive decay problems with half-life. The carbon-14 decay formula C(t) = C₀(1/2)^(t/5730) shows that the sample halves every 5730 years. To find when C(t) = 0.30C₀, we set up 0.30C₀ = C₀(1/2)^(t/5730) and solve for t. Dividing by C₀ gives 0.30 = (1/2)^(t/5730), then taking log base 2 of both sides yields log₂(0.30) = t/5730, so t = 5730·log₂(0.30) = 5730·log₂(1/0.30) years. Choice A is correct because log₂(0.30) = -log₂(1/0.30), and the negative sign makes the time positive since we're looking at decay. Choice C incorrectly keeps log₂(0.30) which would give a negative time, while Choice B uses the wrong logarithm base. To help students: Practice with half-life problems and emphasize that log₂(a) = -log₂(1/a) for positive a.

This question tests AP Precalculus skills: solving exponential growth equations using logarithms in biological contexts. Logarithms transform the exponential equation into a form where we can isolate the time variable. We need to solve $800(1.25)^t$ = 5000, which simplifies to $(1.25)^t$ = 5000/800 = 6.25. Choice A is correct because taking logarithms gives t·log(1.25) = log(6.25), so t = log(5000/800)/log(1.25). Choice B incorrectly inverts the fraction, while Choice C uses subtraction instead of division for the bacteria counts. To help students: Emphasize the importance of simplifying the exponential equation first. Practice recognizing that bacterial growth problems typically involve finding ratios, not differences.

This question tests AP Precalculus skills: understanding and applying natural logarithms to solve exponential growth problems with base e. The culture growth formula N(t) = 300e^(0.18t) represents continuous exponential growth. To find when N(t) = 1200, we set up 1200 = 300e^(0.18t) and solve for t. Dividing by 300 gives 4 = e^(0.18t), then taking natural logarithm of both sides yields ln(4) = 0.18t, so t = ln(4)/0.18. Choice A is correct because it properly applies the natural logarithm to isolate t from the exponential expression. Choice C incorrectly inverts the fraction, while Choice B attempts to use logarithms of the population values directly without proper manipulation. To help students: Emphasize that when dealing with base e, using natural logarithm (ln) simplifies calculations since $ln(e^x$) = x.

This question tests AP Precalculus skills: converting between logarithmic and exponential forms in real-world applications. The decibel scale uses logarithms to measure sound intensity on a more manageable scale. Given L = 10log₁₀(I/I₀) = 70, we need to solve for I/I₀ by reversing the logarithm. Choice A is correct because dividing by 10 gives log₁₀(I/I₀) = 7, which means I/I₀ = $10^7$ by the definition of logarithm. Choice B incorrectly multiplies 7 by 10, while Choice C raises 10 to the 70th power instead of the 7th. To help students: Practice converting between log₁₀(x) = y and x = $10^y$ forms. Emphasize that in the decibel formula, we must first isolate the logarithm by dividing by 10.

This question tests AP Precalculus skills: using logarithms to solve exponential growth equations in population modeling. The population model P(t) = $80,000(1.04)^t$ represents 4% annual growth from an initial population of 80,000. To find when the population doubles to 160,000, we solve 160,000 = $80,000(1.04)^t$. Choice A is correct because dividing by 80,000 gives 2 = $(1.04)^t$, and taking logarithms yields log(2) = t·log(1.04), so t = log(2)/log(1.04). Choice B incorrectly inverts the fraction, which would give the reciprocal of the correct answer. To help students: Emphasize the pattern that when solving $b^x$ = a, we get x = log(a)/log(b). Practice identifying what goes in the numerator (the result) versus denominator (the base).

This question tests AP Precalculus skills: using natural logarithms to solve exponential decay inequalities. The decay formula M(t) = 50e^(-0.3t) uses the natural exponential base e with decay constant -0.3. To find when M(t) < 10, we solve 50e^(-0.3t) < 10. Choice A is correct because dividing by 50 gives e^(-0.3t) < 10/50, taking natural logarithms yields -0.3t < ln(10/50), and dividing by -0.3 (flipping the inequality) gives t > ln(10/50)/(-0.3). Choice C incorrectly uses ln(50/10) instead of ln(10/50), missing that we need the ratio of final to initial mass. To help students: Remember that dividing by a negative number reverses the inequality direction. Practice recognizing when to use natural logarithms (ln) versus common logarithms (log).