Logarithms and Logarithmic Functions: Algebra 2 Math Test Answer Key – PDF Download

Question 1: Convert forms (2 points)

1. \(\log_{5} 125=3\)
2. \(9^{1/2}=3\)

Scoring: 1 point for each correct conversion.

Question 2: Evaluate logs (2 points)

1. \(3^{5}=243\), so 5
2. \(10^{-4}=0.0001\), so -4

Scoring: 1 point for each correct value.

Question 3: Graph of a logarithm (3 points)

1. \(x=-3\)
2. Domain \(x > -3\), range all real numbers
3. \(\log_{2}(x+3)=1\), so \(x+3=2\) and \(x=-1\)

Scoring: 1 point each for the asymptote, the domain and range together, and the x-intercept.

Question 4: Domain (2 points)

1. \(4x-12 > 0\), so \(x > 3\)
2. \(7-x > 0\), so \(x < 7\)

Scoring: 1 point for each correct domain.

Question 5: Properties of logs (2 points)

1. \(\log_{2} 8+2\log_{2} x-\log_{2} y=\)\(3+2\log_{2} x-\log_{2} y\)
2. \(\log 25+\log 4=\log 100=\)2

Scoring: 1 point for each correct result.

Question 6: Change of base (2 points)

\(\log_{7} 30=\frac{\ln 30}{\ln 7}\approx\frac{3.4012}{1.9459}\approx\)1.748

Scoring: 1 point for a correct quotient of logs, 1 point for the value 1.748.

Question 7: Inverse function (3 points)

1. \(x=3^{y}+4\), so \(3^{y}=x-4\) and \(f^{-1}(x)=\log_{3}(x-4)\)
2. \(\log_{3} 27=\)3

Scoring: (a) 1 point for swapping and isolating the power, 1 point for the correct inverse; (b) 1 point.

Question 8: Acidity (2 points)

1. \(-\log 0.0063\approx\)2.20
2. \([H^{+}]=10^{-8.4}\approx\)\(3.98\times 10^{-9}\) mol/L

Scoring: 1 point for each correct answer.

Question 9: Sound intensity (2 points)

1. \(\log\frac{I}{I_{0}}=7\), so the ratio is \(10^{7}\)
2. \(L=10\log(5\times 10^{8})=10(8+0.699)\approx\)87.0 dB

Scoring: 1 point for each correct answer.

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