Solving Exponential and Logarithmic Equations: Algebra 2 Math Test Answer Key – PDF Download

Algebra 2 Exponential Test Answer Key: questions 1–5

Question 1: Common base (2 points)

1. \(3^{2x-1}=3^{4}\), \(2x-1=4\), so \(x=\frac{5}{2}\)
2. \(5^{x+2}=5^{-2}\), \(x+2=-2\), so \(x=-4\)

Scoring: 1 point for each correct solution.

Question 2: Solve with logarithms (2 points)

1. \(x=\frac{\ln 90}{\ln 6}\approx\frac{4.4998}{1.7918}\approx\)2.511
2. \(0.4x=\ln 14\approx 2.6391\), so \(x\approx\)6.598

Scoring: 1 point for each correct decimal.

Question 3: Logarithmic equations (2 points)

1. \(2x+1=3^{4}=81\), so \(x=40\)
2. \(x-2=e^{1.5}\approx 4.4817\), so \(x\approx 6.482\)

Scoring: 1 point for each correct solution.

Question 4: Combine logarithms (3 points)

1. \(\log_{2}(x(x+2))=3\), so \(x(x+2)=8\).
2. \(x^{2}+2x-8=0\), \((x+4)(x-2)=0\), so \(x=2\) (check: \(1+2=3\)).
3. \(x=-4\) is extraneous because \(\log_{2}(-4)\) is undefined.

Scoring: 1 point for each part.

Question 5: Quotient of logarithms (2 points)

\(\log_{5}\frac{3x+2}{x}=1\), so \(\frac{3x+2}{x}=5\), \(3x+2=5x\), and \(x=1\). Check: \(\log_{5} 5-\log_{5} 1=1\).

Scoring: 1 point for combining into one log and rewriting; 1 point for the solution.

Algebra 2 Exponential Test Answer Key: questions 6–9

Question 6: Quadratic form (2 points)

Let \(u=3^{x}\): \(u^{2}-10u+9=0\), \((u-1)(u-9)=0\). So \(3^{x}=1\) gives \(x=0\) and \(3^{x}=9\) gives \(x=2\). \(x=0\) or \(x=2\)

Scoring: 1 point for the factored quadratic in u; 1 point for both values of x.

Question 7: Compound interest (3 points)

1. \(2500(1.003)^{12t}=4000\)
2. \(1.003^{12t}=1.6\), \(12t=\frac{\ln 1.6}{\ln 1.003}\approx\frac{0.4700}{0.002996}\approx 156.9\), so \(t\approx 13.08\) years

Scoring: (a) 1 point; (b) 1 point for isolating the power and logging, 1 point for the value.

Question 8: Half-life (2 points)

1. \((\frac{1}{2})^{t/8}=\frac{1}{8}\), so \(\frac{t}{8}=3\) and \(t=24\) days
2. \((\frac{1}{2})^{t/8}=\frac{1}{3}\), \(\frac{t}{8}=\log_{2} 3\approx 1.585\), so \(t\approx 12.68\) days

Scoring: 1 point for each correct time.

Question 9: Equivalent forms (2 points)

1. \(1.06^{t}=e^{t\ln 1.06}\), so \(k=\ln 1.06\approx 0.0583\)
2. \(1.06^{1/12}\approx 1.004868\), so 0.487% per month

Scoring: 1 point for each part.

Scoring summary

How to use this algebra 2 exponential test answer key

Use this algebra 2 exponential test answer key to score the test fairly and to see which skills need more practice.

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