Logarithms Test Answer Key with Scoring – Algebra 2

Question 1 Convert between forms (2 points)

1. log₇ 49 = 2.
2. 10⁻² = 0.01.

Scoring: 1 point for each correct form.

Question 2 Evaluate logarithms (3 points)

1. 4³ = 64, so 3.
2. 9^1/2 = 3, so 1/2.
3. 7 + 3 = 10.

Scoring: 1 point for each correct value.

Question 3 Read a logarithmic graph (3 points)

1. x = −2, where x + 2 = 0.
2. x > −2.
3. log₂(6 + 2) − 1 = log₂ 8 − 1 = 3 − 1 = 2. Verified.

Scoring: 1 point for each part.

Question 4 Expand a logarithm (2 points)

log₂ 8 + 2 log₂ p − log₂ q = 3 + 2 log₂ p − log₂ q.

Scoring: 1 point for the product and quotient rules, 1 point for the power rule and the value 3.

Question 5 Condense and simplify (2 points)

log₆(12 · 3) = log₆ 36 = 2, since 6² = 36.

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

Question 6 Change of base (2 points)

log₇ 90 = ln 90 ÷ ln 7 ≈ 4.4998 ÷ 1.9459 ≈ 2.31.

Scoring: 1 point for the quotient setup, 1 point for the value.

Question 7 pH of a solution (2 points)

pH = −log(4 × 10⁻⁹) = 9 − log 4 ≈ 9 − 0.602 = 8.40. Since 8.40 > 7, the solution is basic.

Scoring: 1 point for the pH value, 1 point for the conclusion.

Question 8 Inverse from a graph (2 points)

Swap coordinates: (2, 1), (4, 2), (8, 3). The inverse is y = log₂ x.

Scoring: 1 point for the three points, 1 point for the equation.

Question 9 Decibels (2 points)

I/I₀ = 8 × 10⁶. log(8 × 10⁶) = 0.9031 + 6 = 6.9031, so L = 69.031. ≈ 69.0 dB.

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