## Precalculus: Mathematics for Calculus, 7th Edition

a.) -3 b.) $\frac{1}{2}$ c.) -1
$Evaluate$ $the$ $expression:$ a.) $\log_3 (\frac{1}{27})$ b.) $\log_{10} \sqrt{10}$ c.) $\log_5 0.2$ a.) $\log_3 (\frac{1}{27})$ Rewrite $\frac{1}{27}$ as $3^{-3}$ [Note: $3^{-3} = \frac{1}{3^3} = \frac{1}{3\times3\times3} = \frac{1}{27}$] $\log_3 3^{-3}$ Use the Third Property of Logarithms: $\log_a a^x = x$ $\log_3 3^{-3} = -3$ b.) $\log_{10} \sqrt{10}$ Rewrite the root to exponential form: $\sqrt{x} = x^{\frac{1}{2}}$] $\log_{10} 10^{\frac{1}{2}}$ Use the Third Property of Logarithms: $\log_a a^x = x$ $\log_{10} 10^{\frac{1}{2}} = \frac{1}{2}$ c.) $\log_5 0.2$ Rewrite 0.2 as $5^{-1}$ [Note: $5^{-1} = \frac{1}{5} = 0.2$] $\log_5 5^{-1}$ Use the Third Property of Logarithms: $\log_a a^x = x$ $\log_5 5^{-1} = -1$