1.
Solve the radical equation: \(\sqrt{x+4}=\sqrt{3x-1}\).
Restrictions: \(x+4\ge0\) and \(3x-1\ge0\), so \(x\ge\frac13\). Square both sides: \(x+4=3x-1\). Then \(2x=5\), so \(x=\frac52\). This satisfies the restriction. Answer: B
2.
Which restriction of \(\log_b a=c\) is not being obeyed in the logarithmic expression \(\log_5(-8)=-1.292\)?
For \(\log_b a=c\), the argument must satisfy \(a>0\). In \(\log_5(-8)\), the argument is \(-8\), which is not positive. Therefore, \(a>0\) is not obeyed. Answer: D
3.
Solve exactly for \(x\): \(3^x=18\).
Take logarithms of both sides: \(\log(3^x)=\log18\). Then \(x\log3=\log18\), so \(x=\frac{\log18}{\log3}\). Answer: A
4.
Evaluate: \(\log_{12}8000\).
Use change of base: \(\log_{12}8000=\frac{\log 8000}{\log 12}\approx3.62\). Answer: C
5.
Write as a single logarithm: \(5\log_3 A+\log_3 B-2\).
Use the power rule: \(5\log_3A=\log_3(A^5)\). Also, \(2=\log_3(9)\). Therefore, \(5\log_3A+\log_3B-2=\log_3(A^5)+\log_3B-\log_3(9)=\log_3\left(\frac{A^5B}{9}\right)\). Answer: D
6.
The World's largest sphere is located in Astana, Kazakhstan. The Nur Alem, built for the 2017 Future Science Expo, has a diameter of \(80 ext{ m}\). What is its volume? Use \(V=rac{4}{3}\pi r^3\).
The diameter is \(80\text{ m}\), so the radius is \(40\text{ m}\). Use \(V=\frac{4}{3}\pi r^3\): \(V=\frac{4}{3}\pi(40)^3\approx268083\text{ m}^3\). Answer: D
7.
Solve: \(\log_3 x+\log_3(x-6)=3\).
Domain: \(x>0\) and \(x-6>0\), so \(x>6\). Combine logs: \(\log_3[x(x-6)]=3\). Convert to exponential form: \(x(x-6)=3^3=27\). Then \(x^2-6x-27=0=(x-9)(x+3)\). Thus \(x=9\) or \(x=-3\), but only \(x=9\) satisfies the domain. Answer: B
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