Exponential and Logarithmic Functions and Models: Precalculus Math Test Answer Key – PDF Download

Precalculus Exponential Log Test Answer Key: questions 1–5

Question 1: Logarithm basics (2 points)

1. \(\log_{7} 343=3\)
2. \((\frac{1}{3})^{y}=9\) gives \(3^{-y}=3^{2}\), so \(y=-2\)

Scoring: 1 point for each correct part.

Question 2: Laws of logarithms (2 points)

1. \(3\ln x+\frac{1}{2}\ln y-2\ln z\)
2. \(\log a^{2}+\log b^{3}-\log c=\)\(\log\frac{a^{2}b^{3}}{c}\)

Scoring: 1 point for each part: the power law, the quotient law and the product law all applied correctly.

Question 3: Inverse function (2 points)

1. Solve \(x=3\ln (y-1)+2\): \(\frac{x-2}{3}=\ln (y-1)\), so \(f^{-1}(x)=e^{(x-2)/3}+1\)
2. The range of \(f\) is all real numbers, so the domain of \(f^{-1}\) is all real numbers.

Scoring: 1 point for the inverse (0.5 for isolating the logarithm, 0.5 for exponentiating), 1 point for the domain.

Question 4: Graph and inverse (2 points)

1. The asymptote \(y=-2\) and the point \((1,0)\) fit \(f(x)=2^{x}-2\): \(2^{1}-2=0\), \(2^{3}-2=6\). Answer: \(f(x)=2^{x}-2\)
2. \(x=2^{y}-2\) gives \(2^{y}=x+2\), so \(f^{-1}(x)=\log_{2}(x+2)\) with asymptote \(x=-2\)

Scoring: 1 point for the equation of f (0.5 if the shift is right but the base is wrong); 1 point for the inverse (0.5) and its asymptote (0.5).

Question 5: Exponential equations (2 points)

1. \(e^{0.3t}=5\), so \(t=\frac{\ln 5}{0.3}\approx\) 5.36 hours
2. \((x+1)\ln 2=x\ln 3\), so \(x(\ln 3-\ln 2)=\ln 2\) and \(x=\frac{\ln 2}{\ln 3-\ln 2}\approx\) 1.710

Scoring: 1 point per part: 0.5 for the correct setup with logarithms, 0.5 for the correct rounded value.

Precalculus Exponential Log Test Answer Key: questions 6–9

Question 6: Logarithmic equation (2 points)

\(\log_{3}\big((x+2)(x-4)\big)=3\), so \((x+2)(x-4)=27\).

\(x^{2}-2x-35=0\) gives \((x-7)(x+5)=0\).

For \(x=-5\), the argument \(x-4=-9\) is negative, so it is rejected. Check \(x=7\): \(\log_{3}9+\log_{3}3=2+1=3\). The solution is \(x=7\).

Scoring: 1 point for combining into a single equation and solving the quadratic; 1 point for rejecting x = -5 with a correct reason and stating x = 7.

Question 7: Continuous growth (2 points)

1. \(P(10)=12000e^{0.24}\approx 12000(1.27125)\approx\) 15,255 people
2. \(e^{0.024t}=\frac{5}{3}\), so \(t=\frac{\ln (5/3)}{0.024}\approx\) 21.28 years

Scoring: 1 point per part: 0.5 for the correct substitution or setup, 0.5 for the correct value.

Question 8: Logistic model (3 points)

1. \(P(0)=\frac{800}{1+7}=\)100 fish
2. As \(t\) grows, \(e^{-0.4t}\to 0\), so \(P\to 800\): 800 fish
3. \(1+7e^{-0.4t}=2\), so \(e^{-0.4t}=\frac{1}{7}\) and \(t=\frac{\ln 7}{0.4}\approx\) 4.86 years

Scoring: 1 point per part. In part (c), award 0.5 for setting the denominator equal to 2 and 0.5 for the final value.

Question 9: Logarithmic scales (3 points)

1. \(10^{6.8-5.6}=10^{1.2}\approx\) 15.8 times
2. \(\text{pH}=-\log (2.5\times 10^{-4})\approx\) 3.60
3. \(10^{(95-80)/10}=10^{1.5}\approx\) 31.6 times

Scoring: 1 point per part: 0.5 for the correct power of 10 or logarithm, 0.5 for the correct rounded value.

Scoring summary

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