Answer
a.) -3
b.) $\frac{1}{2}$
c.) -1
Work Step by Step
$Evaluate$ $the$ $expression:$
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: $\log_a a^x = x$
$\log_5 5^{-1} = -1$