How to calculate log 5 </msub> &#x2061;<!-- ⁡ --> tan &#x2061;<!-- ⁡ -->

How to calculate ${\mathrm{log}}_{5}\mathrm{tan}\left({36}^{\circ }\right)+{\mathrm{log}}_{5}\mathrm{tan}\left({54}^{\circ }\right)$ without a calculator?
You can still ask an expert for help

Want to know more about Trigonometry?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Mateo Carson
Hint: ${36}^{\circ }+{54}^{\circ }=90$ so
$\mathrm{tan}\left({54}^{\circ }\right)=\mathrm{cot}\left({36}^{\circ }\right)$
${\mathrm{log}}_{5}\mathrm{tan}\left({36}^{\circ }\right)+{\mathrm{log}}_{5}\mathrm{tan}\left({54}^{\circ }\right)={\mathrm{log}}_{5}\left(\mathrm{tan}\left({36}^{\circ }\right)\mathrm{tan}\left({54}^{\circ }\right)\right)=\phantom{\rule{0ex}{0ex}}{\mathrm{log}}_{5}\left(\mathrm{tan}\left({36}^{\circ }\right)\mathrm{cot}\left({36}^{\circ }\right)\right)={\mathrm{log}}_{5}\left(1\right)=0$