(a) Given that (3 times 9^{1 + x} = 27^{-x}), find x.
(b) Evaluate (log_{10} sqrt{35} + log_{10} sqrt{2} – log_{10} sqrt{7})
Explanation
(a) (3 times 9^{1 + x} = 27^{-x})
(3 times (3^{2})^{1 + x} = (3^{3})^{-x})
(3 times 3^{2 + 2x} = 3^{-3x} )
(3^{2x + 2 + 1} = 3^{-3x} implies 3^{2x + 3} = 3^{-3x})
(implies 2x + 3 = – 3x)
(-5x = 3 implies x = frac{-3}{5})
(b) (log_{10} sqrt{35} + log_{10} sqrt{2} – log_{10} sqrt{7})
= (log_{10} (frac{sqrt{35} times sqrt{2}}{sqrt{7}})
= (log_{10} (frac{sqrt{35 times 2}{sqrt{7}})
= (log_{10} sqrt{10})
= (log_{10} 10^{frac{1}{2}})
= (frac{1}{2} log_{10} 10 = frac{1}{2})