1 Evaluating logarithms / 4 pts
Evaluate without a calculator: (a) \( \log_2 64 \) (b) \( \log_5\dfrac1{125} \) (c) \( \ln e^{-2} \) (d) \( 10^{\log 4.5} \).
2 Properties of logarithms / 3 pts
(a) Expand \( \log_3\dfrac{27x^{2}}{\sqrt y} \) for \( x,y\gt0 \). (b) Write \( 2\ln3+\ln5-\ln15 \) as a single logarithm and simplify.
3 Exponential equations (calculator allowed for part b) / 4 pts
Solve: (a) \( 5^{x-2}=125 \) (b) \( 3\cdot4^{x}=51 \), rounded to four decimals (c) \( e^{2x}-5e^{x}+6=0 \).
4 Logarithmic equations / 3 pts
Solve: (a) \( \log_3(2x-5)=2 \) (b) \( \ln x+\ln(x+4)=\ln12 \).
5 Compound interest (calculator allowed) / 4 pts
You deposit $6,000 at an annual rate of 5.2%. (a) Find the balance after 7 years if interest is compounded quarterly. (b) Find the balance after 7 years with continuous compounding. (c) With continuous compounding, how many years until the balance reaches $9,000?
6 Radioactive decay (calculator allowed) / 2 pts
A 640 mg sample of an isotope has a half-life of 12 days. (a) How many mg remain after 30 days? (b) After how many days will 40 mg remain?
Test yourself: quick challenge for Grade 12
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