Precalculus Exponential and Logarithmic Equations Test Solutions and Scoring (PDF)

(a) \(81=3^{4}\), so \(x+2=4\). (b) \(x=\frac{\ln50}{\ln7}\approx 2.010\).

(a) \(x=2\); (b) \(x\approx 2.010\)

Scoring: 1 point each.

\(3x=\ln20\), so \(x=\frac{\ln20}{3}\approx 0.999\).

\(x\approx 0.999\)

Scoring: 1 point for the log step, 1 point for the value.

\(2^{x}=14\), so \(x=\frac{\ln14}{\ln2}\approx 3.807\).

\(x\approx 3.807\)

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

\(2x-1=3^{2}=9\), so \(x=5\).

\(x=5\)

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

(a) \(\log\left(x(x+3)\right)=1\). (b) \(x^{2}+3x=10\), \((x+5)(x-2)=0\). (c) \(x=-5\) makes \(\log(-5)\) undefined, so reject it; \(\log2+\log5=\log10=1\).

\(x=2\)

Scoring: 1 point for each part.

\(x(x-2)=15\), so \(x^{2}-2x-15=0\) and \((x-5)(x+3)=0\). Reject \(x=-3\).

\(x=5\)

Scoring: 1 point for the equation, 1 point for rejecting -3.

\(1.04^{t}=1.6\), \(t=\frac{\ln1.6}{\ln1.04}\approx 11.98\).

\(11.98\) years

Scoring: 1 point for the equation, 1 point for t.

(a) \(800e^{-0.5}\approx 485.2\). (b) \(e^{-0.05t}=\frac12\), \(t=\frac{\ln2}{0.05}\approx 13.86\). (c) \(e^{-0.05t}=\frac18\), \(t=\frac{\ln8}{0.05}\approx 41.59\).

\(485.2\text{ g}\); \(13.86\) days; \(41.59\) days

Scoring: 1 point for each part.

\((x-1)\ln3=x\ln2\), so \(x(\ln3-\ln2)=\ln3\) and \(x=\frac{\ln3}{\ln1.5}\approx 2.710\).

\(x\approx 2.710\)

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