Precalculus Logarithms and Their Properties Test Solutions and Scoring (PDF)

(a) \(\log_{3}81=4\). (b) \(2^{-4}=\frac{1}{16}\).

(a) \(\log_{3}81=4\); (b) \(2^{-4}=\frac1{16}\)

Scoring: 1 point each.

\(5^{3}=125\). \(\left(\frac13\right)^{-2}=9\).

\(3\) and \(-2\)

Scoring: 1 point each.

\(5-x>0\) gives \(x< 5\). The asymptote is where \(5-x=0\).

Domain \((-\infty,5)\); asymptote \(x=5\)

Scoring: 1 point each.

\(\log x^{2}+\log y^{1/2}-\log z\).

\(2\log x+\frac12\log y-\log z\)

Scoring: 1 point for the product and quotient structure, 1 point for the powers.

(a) \(x=0\). (b) Domain \((0,\infty)\); range all real numbers. (c) \(y=3^{x}\).

\(x=0\); \((0,\infty)\), all reals; \(y=3^{x}\)

Scoring: 1 point for each part.

\(\ln x^{3}-\ln y^{2}+\ln5=\ln\frac{5x^{3}}{y^{2}}\).

\(\ln\frac{5x^{3}}{y^{2}}\)

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

\(\frac{\ln 50}{\ln 7}=\frac{3.9120}{1.9459}\approx 2.010\).

\(2.010\)

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

(a) \(-\log(2.5\times10^{-5})\approx 4.60\), about 4.6. (b) \(10^{-8}\). (c) \(10^{-3}\div10^{-5}=100\).

\(4.6\); \(10^{-8}\); \(100\) times

Scoring: 1 point for each part.

\(\log_{b}10=0.69+1.61=2.30\). \(\log_{b}2.5=\log_{b}\frac52=1.61-0.69=0.92\).

\(2.30\) and \(0.92\)

Scoring: 1 point each.