Answer
a.) 3
b.) $\frac{1}{2}$
c.) $\frac{1}{4}$
Work Step by Step
$Evaluate$ $the$ $expression:$
a.) $\log_5 125$
b.) $\log_{49} 7$
c.) $\log_9 \sqrt 3$
a.) $\log_5 125$
Rewrite 125 as $5^3$ [Note: $5^3 = 5\times5\times5 = 125$]
$\log_5 5^3$
Use the Third Property of Logarithms: $\log_a a^x = x$
$\log_5 5^3 = 3$
b.) $\log_{49} 7$
Rewrite 7 as $49^{\frac{1}{2}}$ [Note: $49^{\frac{1}{2}} = \sqrt{49} = 7$]
$\log_{49} 49^{\frac{1}{2}}$
Use the Third Property of Logarithms: $\log_a a^x = x$
$\log_{49} 49^{\frac{1}{2}} = \frac{1}{2}$
c.) $\log_9 \sqrt 3$
Rewrite 3 as $9^{\frac{1}{2}}$ [Note: $9^{\frac{1}{2}} = \sqrt 9 = 3$]
$\log_9 \sqrt{9^{\frac{1}{2}}}$
Rewrite the root to exponential form
$\log_9 (9^{\frac{1}{2}})^{\frac{1}{2}} \rightarrow \log_9 9^{\frac{1}{2}\times\frac{1}{2}}$
$\log_9 9^{\frac{1}{4}}$
Use the Third Property of Logarithms: $\log_a a^x = x$
$\log_9 9^{\frac{1}{4}} = \frac{1}{4}$
