Paul's Online Notes
Paul's Online Notes
Home / Algebra / Exponential and Logarithm Functions / 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 6.2 : Logarithm Functions

9. Without using a calculator determine the exact value of \({\log _2} \displaystyle \frac{1}{8}\).

Show All Steps Hide All Steps

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

Converting the logarithm to exponential form gives,

\[{\log _2}\frac{1}{8} = ?\hspace{0.25in} \Rightarrow \hspace{0.25in}{2^?} = \frac{1}{8}\] Show Step 2

Now, we know that if we raise an integer to a negative exponent we’ll get a fraction and so we must have a negative exponent and then we know that \({2^3} = 8\). Therefore, we can see that \({2^{ - 3}} = \frac{1}{8}\) and so we must have,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{{{\log }_2}\frac{1}{8} = - 3}}\]