Paul's Online Notes
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

14. Use the change of base formula and a calculator to find the value of \({\log _{\frac{2}{3}}}53\).

Show Solution

We can use either the natural logarithm or the common logarithm to do this so we’ll do both.

\[{\log _{\frac{2}{3}}}53 = \frac{{\ln 53}}{{\ln \frac{2}{3}}} = \frac{{3.97029191}}{{ - 0.40546511}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 9.79194469}}\] \[{\log _{\frac{2}{3}}}53 = \frac{{\log 53}}{{\log \frac{2}{3}}} = \frac{{1.72427587}}{{ - 0.17609126}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 9.79194469}}\]

So, as we noted at the start it doesn’t matter which logarithm we use we’ll get the same answer in the end.