Five Fives

Extending the idea of Four Fours, where the intention is to write each integer as a combination of 4 4's (any mathematical symbol is allowed except for digits that are not 4), we can create the game of Five Fives.

It is also useful to build a table of different combinations of 5 to help out: Five Fives/Table of 5s

${\displaystyle 1=\left({\dfrac {\ln {\left({\dfrac {5}{\sqrt {5}}}\right)}}{\ln {5}}}\right)+\left({\dfrac {\ln {\sqrt {5}}}{\ln {5}}}\right)}$

${\displaystyle 1={\dfrac {5\times 5\times 5-5}{5!}}}$

${\displaystyle 1=\left({\dfrac {5+5}{5+5}}\right)^{5}}$

${\displaystyle 2={\dfrac {5\ln {5}}{\ln {\left({\dfrac {5^{5}}{\sqrt {5}}}\right)}}}}$

${\displaystyle 3={\dfrac {\log _{5}{(5!+5)^{5}}}{5}}}$

${\displaystyle 4=5-{\dfrac {5^{5}}{5^{5}}}}$

${\displaystyle 5={\dfrac {{\sqrt {5}}\times {\sqrt {5}}^{5}}{5\times 5}}}$

${\displaystyle 5=5\times {\dfrac {5}{5}}\times {\dfrac {5}{5}}}$

${\displaystyle 6=5+{\dfrac {5\times 5}{5\times 5}}}$

${\displaystyle 7=5+{\dfrac {5}{5}}+{\dfrac {5}{5}}}$

${\displaystyle 8=5+{\dfrac {5+5+5}{5}}}$

${\displaystyle 9={\sqrt {5}}{\sqrt {5}}+5-{\dfrac {5}{5}}}$

${\displaystyle 10={\dfrac {5\times 5+5\times 5}{5}}}$

${\displaystyle 11={\dfrac {5\times 5+5}{5}}+5}$

${\displaystyle 12=5+5+{\dfrac {5+5}{5}}}$

${\displaystyle 13=5+5+5-{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}}$

${\displaystyle 14=5+5+5-{\dfrac {5}{5}}}$

${\displaystyle 15=\left({\dfrac {5+5}{5}}\right)\times 5+5}$

${\displaystyle 16=5+5+5+{\dfrac {5}{5}}}$

${\displaystyle 17=5+5+5+{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}}$

${\displaystyle 18=5\times 5-5-{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}}$

${\displaystyle 19=5\times 5-5-{\dfrac {5}{5}}}$

${\displaystyle 20={\dfrac {5}{5}}\left(5\times 5-5\right)}$

${\displaystyle 21={\dfrac {\sqrt {5^{5}}}{\sqrt {5}}}-{\dfrac {\ln {5}}{\ln {\left({\sqrt {\sqrt {5}}}\right)}}}}$

${\displaystyle 22=(5\times 5)-\log _{5}{\left(5!+5\right)}}$

${\displaystyle 23={\dfrac {5!}{5}}-{\dfrac {\sqrt {5\times 5}}{5}}}$

${\displaystyle 24=5\times 5\times \left({\dfrac {5!+5}{5!}}\right)}$

${\displaystyle 25={\dfrac {5^{5}}{5}}-5\times 5!}$

${\displaystyle 26=5\times 5+{\dfrac {\sqrt {5\times 5}}{5}}}$

${\displaystyle 27={\dfrac {5!}{5}}+5-{\dfrac {\ln {5}}{\ln {\sqrt {5}}}}}$

${\displaystyle 28={\dfrac {5!}{5}}+5-{\dfrac {5}{5}}}$

${\displaystyle 29=5\times 5+5-{\dfrac {5}{5}}}$

${\displaystyle 30=5\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}+{\dfrac {\ln {5}}{\ln {\sqrt {\sqrt {5}}}}}\right)}$

${\displaystyle 31=5\times 5+{\dfrac {5}{5}}}$

${\displaystyle 32=\left({\dfrac {\ln {5}}{\ln {\dfrac {5}{\sqrt {5}}}}}\right)^{\sqrt {5\times 5}}}$

${\displaystyle 33=\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5}+{\dfrac {5}{5}}}$

${\displaystyle 34=5\times 5+5+{\dfrac {\ln {5}}{\ln {\sqrt {\sqrt {5}}}}}}$

$\displaystyle 35 = \dfrac{55}{5} + \dfrac{5!}{5}$

${\displaystyle 36={\dfrac {5^{5}-5}{5!}}+5+5}$

${\displaystyle 37={\dfrac {5!}{5+5}}+5\times 5}$

${\displaystyle 38=\left({\dfrac {5!}{5}}-5\right)\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)}$

${\displaystyle 39={\dfrac {5!}{5}}+5+5+5}$

${\displaystyle 40=55-5-5-5}$

${\displaystyle 41=\phi ^{5}-{\dfrac {1}{\phi ^{5}}}+5\times 5+5}$

(where ${\displaystyle \phi ={\dfrac {1+{\sqrt {5}}}{2}}}$ is the Golden Ratio)

${\displaystyle 42=5+5+\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5}}$

${\displaystyle 43=55-{\dfrac {5!}{5+5}}}$

${\displaystyle 44=\left(5-{\dfrac {5}{5}}\right)\left(\phi ^{5}-{\dfrac {1}{\phi ^{5}}}\right)}$

${\displaystyle 45=55-{\sqrt {5}}\left({\sqrt {5}}+{\sqrt {5}}\right)}$

${\displaystyle 46=55-\left(5+5\right)+\left(\ln {e}\right)^{5}}$