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

How to use this precalculus exponential log test answer key

Use this precalculus exponential log test answer key to score the test fairly and to see which skills need more practice.

Printable PDF · Test Answer Key

Precalculus Exponential and Logarithmic Functions and Models test answer key

• 📄 US Letter
• 💾 200 KB
• 🖨️ Black and white friendly

Download the PDF

See every chapter on the Precalculus math page.

📘 The Mathovore bookPrecalculus Math Tests with Answer KeysA 45-minute test for every chapter

• 18 chapters in one PDF
• Tests scored out of 20 points
• Answer keys with scoring guides
• Perfect to prepare for exams

See the book →