Algebra 2 Logarithms Test Solutions and Scoring (PDF)

1. \(\log_{3}(81) = 4\)
2. \(5^{3} = 125\)

(a) \(\log_{3}(81) = 4\); (b) \(5^{3} = 125\)

Scoring: 1 point for each conversion.

1. \(2^{5} = 32\)

2. \(7^{-1} = \frac{1}{7}\)

(a) 5; (b) -1

Scoring: 1 point for each part.

1. \(\log_{2}(8) + \log_{2}(x^{3}) = 3 + 3\log_{2}(x)\)

2. \(\ln(6 \cdot 5) = \ln 30\)

(a) \(3 + 3\log_{2}(x)\); (b) \(\ln 30\)

Scoring: Part (a): 1 point for the product rule, 1 point for the power rule. Part (b): 1 point.

\(x = \log_{4}(50) = \frac{\ln 50}{\ln 4} = \frac{3.9120}{1.3863} \approx 2.82\).

\(x = \frac{\ln 50}{\ln 4} \approx 2.82\)

Scoring: 1 point for taking logs, 1 point for the exact answer, 1 point for the decimal.

\(2x – 1 = 3^{2} = 9\), so \(x = 5\).

\(x = 5\)

Scoring: 1 point for the exponential form, 1 point for the solution.

\(\frac{\ln 40}{\ln 3} \approx \frac{3.6889}{1.0986} \approx 3.36\).

About 3.36

Scoring: 1 point for the formula, 1 point for the value.

\(t = \frac{\ln 2}{\ln 1.04} \approx 17.67\). The balance passes 2 between 17 and 18 years.

About 17.7 years

Scoring: 1 point for the equation and formula, 1 point for the value.

Only positive inputs have a logarithm. Outputs take every real value.

Domain x > 0; range all real numbers; asymptote x = 0

Scoring: 1 point for the domain and asymptote, 1 point for the range.

At \(x = 10\): \(2\log 10 = 2\) but \(\log 20 \approx 1.301\). The values differ.

By the power rule, \(2\log x = \log x^{2}\).

2 log x = log(x²)

Scoring: 1 point for the counterexample, 1 point for the correct rewrite.