Test solutions with the detailed point scale. Add up your points and spot what to review.
1 Conversions and values / 4 pts
- \(\log_5 125=3\). (1 pt)
- \(2^{-4}=\tfrac1{16}\). (1 pt)
- \(9^{1/2}=3\), so \(\log_9 3=\tfrac12\). (1 pt)
- \(7+3=10\). (1 pt)
2 Properties of logarithms / 3 pts
- \(\log_3 27+2\log_3 x-\log_3 y=3+2\log_3 x-\log_3 y\). (2 pts)
- \(\ln 8+\ln 5-\ln 10=\ln\dfrac{40}{10}=\ln 4\). (1 pt)
3 Exponential equations / 3 pts
- \(216=6^3\), so \(x-2=3\) and \(x=5\). (1 pt)
- \(5^x=20\) (1 pt), so \(x=\dfrac{\ln 20}{\ln 5}\approx 1.861\). (1 pt)
4 Logarithmic equations / 4 pts
- \(3x-2=4^3=64\) (1 pt), so \(x=22\). (1 pt)
- \(x(x+6)=16\), so \(x^2+6x-16=0\) (1 pt), i.e. \((x+8)(x-2)=0\). \(x=-8\) is rejected since the logs would be undefined; the solution is \(x=2\) (check: \(\log_2 2+\log_2 8=1+3=4\)). (1 pt)
5 Reading a graph / 3 pts
- Domain \(x\gt 1\); vertical asymptote \(x=1\). (1 pt)
- A\((2,2)\), B\((4,3)\), C\((10,4)\). Check: \(f(4)=\log_3 3+2=3\) and \(f(10)=\log_3 9+2=4\). (1 pt)
- \(f(28)=\log_3 27+2=3+2=5\). (1 pt)
6 Acidity of solutions / 3 pts
- \(\text{pH}=-\log(0.0025)\approx 2.60\). (1.5 pts)
- \([\text{H}^+]=10^{-8.3}\approx 5.0\times10^{-9}\) mol/L. (1.5 pts)
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