Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 6.2 : Logarithm Functions
10. Without using a calculator determine the exact value of \({\log _{\frac{1}{4}}}16\).
Show All Steps Hide All Steps
Hint : Recall that converting a logarithm to exponential form can often help to evaluate these kinds of logarithms.
Converting the logarithm to exponential form gives,
\[{\log _{\frac{1}{4}}}16 = ?\hspace{0.25in} \Rightarrow \hspace{0.25in}{\left( {\frac{1}{4}} \right)^?} = 16\] Show Step 2Now, we know that if we raise an fraction to a power and get an integer out we must have had a negative exponent. Now, we also know that \({4^2} = 16\). Therefore, we can see that \({\left( {\frac{1}{4}} \right)^{ - 2}} = {\left( {\frac{4}{1}} \right)^2} = 16\) and so we must have,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{{{\log }_{\frac{1}{4}}}16 = - 2}}\]