Paul's Online Notes
Home / Calculus I / Review / Logarithm Functions
Show Mobile Notice Show All Notes Hide All Notes
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 views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 1.8 : Logarithm Functions

4. Without using a calculator determine the exact value of $${\log _{\frac{1}{4}}}16$$.

Hint : Recall that converting a logarithm to exponential form can often help to evaluate these kinds of logarithms.
Show Solution

Converting the logarithm to exponential form gives,

${\log _{\frac{1}{4}}}16 = ?\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}{\left( {\frac{1}{4}} \right)^?} = 16$

Now, 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,

$\require{bbox} \bbox[2pt,border:1px solid black]{{{{\log }_{\frac{1}{4}}}16 = - 2}}$