Back to Five Fives

# One 5

Various ways of arranging a single 5 to yield different numbers. (More limited than 4, of course...)

${\displaystyle 5^{\frac {1}{2}}={\sqrt {5}}}$

${\displaystyle 5=5}$

${\displaystyle 120=5!}$

# Two 5s

${\displaystyle 0=\ln {\dfrac {5}{5}}}$

${\displaystyle 1={\dfrac {5}{5}}}$

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

${\displaystyle 4={\dfrac {\ln {5}}{\ln {\left({\sqrt {\sqrt {5}}}\right)}}}}$

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

${\displaystyle 10=5+5}$

${\displaystyle 11=\phi ^{5}-{\dfrac {1}{\phi ^{5}}}}$

${\displaystyle 24={\dfrac {5!}{5}}}$

${\displaystyle 25=5\times 5}$

${\displaystyle 55}$

${\displaystyle 115=5!-5}$

${\displaystyle 125=5!+5}$

${\displaystyle 600=5\times 5!}$

${\displaystyle 3125=5^{5}}$

${\displaystyle 12696403353658275925965100847566516959580321051449436762275840000000000000=55!}$

## Three 5s

${\displaystyle {\dfrac {1}{2}}={\dfrac {\ln {\left({\dfrac {5}{\sqrt {5}}}\right)}}{\ln {(5)}}}}$

${\displaystyle {\dfrac {25}{24}}={\dfrac {5!+5}{5!}}}$

${\displaystyle 1={\dfrac {\sqrt {5\times 5}}{5}}}$

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

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

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

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

${\displaystyle 5={\sqrt[{5}]{5^{5}}}}$

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

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

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

${\displaystyle 10={\sqrt {5}}\left({\sqrt {5}}+{\sqrt {5}}\right)}$

${\displaystyle 11={\dfrac {55}{5}}}$

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

${\displaystyle 15=5+5+5}$

${\displaystyle 16=\phi ^{5}-{\dfrac {1}{\phi ^{5}}}+5}$

${\displaystyle 20=5\times 5-5}$

${\displaystyle 20={\dfrac {5\lg {5}}{\lg {\left({\sqrt {\sqrt {5}}}\right)}}}}$

${\displaystyle 30=5\times 5+5}$

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

${\displaystyle 25={\dfrac {\sqrt {5^{5}}}{\sqrt {5}}}}$

${\displaystyle 60={\dfrac {5!}{\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)}}}$

${\displaystyle 95=5!-5\times 5}$

${\displaystyle 130=5!+5+5}$

${\displaystyle 145=5!+5\times 5}$

${\displaystyle 625={\dfrac {5^{5}}{5}}}$

${\displaystyle 1024=\left({\dfrac {\ln {5}}{\ln {\sqrt {\sqrt {5}}}}}\right)^{5}}$

${\displaystyle 503284375=55^{5}}$

## Four 5s

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

${\displaystyle 1={\dfrac {5^{5}}{5^{5}}}}$

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

${\displaystyle 6={\dfrac {\ln {(5)}}{\ln {({\sqrt {5}})}}}+{\dfrac {\ln {(5)}}{\ln {({\sqrt {\sqrt {5}}})}}}}$

${\displaystyle 7={\sqrt {{\dfrac {5!}{5}}+(5\times 5)}}}$

${\displaystyle 8=\left({\dfrac {\ln {(5)}}{\ln {({\sqrt {5}})}}}\right)\left({\dfrac {\ln {(5)}}{\ln {({\sqrt {\sqrt {5}}})}}}\right)}$

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

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

${\displaystyle 12=\left({\dfrac {5!}{5}}\right)\times \left({\dfrac {\ln {\sqrt {5}}}{\ln {5}}}\right)}$

${\displaystyle 13={\dfrac {5!-55}{5}}}$

${\displaystyle 15=\left(\log _{5}{(5!+5)^{5}}\right)}$

${\displaystyle 20=5+5+5+5}$

${\displaystyle 26={\dfrac {5^{5}-5}{5!}}}$

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

${\displaystyle 35=5\times 5+5+5}$

${\displaystyle 49={\dfrac {5!}{5}}+(5\times 5)}$

${\displaystyle 50=5\times 5+5\times 5}$

${\displaystyle 60=5\left({\dfrac {5!}{5+5}}\right)}$

${\displaystyle 70=5!+5-55}$

${\displaystyle 120=5\times 5\times 5-5}$

${\displaystyle 123=\phi ^{5+5}+{\dfrac {1}{\phi ^{5+5}}}}$

${\displaystyle 130=5\times 5\times 5+5}$

${\displaystyle 243=\left(\log _{5}{(5!+5)}\right)^{5}}$

${\displaystyle (5_{5})!=(10_{5})!=440_{5}=120_{10}}$

(5 base 5 is 10 base 5; 5 base 5 factorial is 440 base 5, or 120 base 10. No surprises here.)

${\displaystyle 505={\dfrac {5^{5}}{5}}-5!}$

${\displaystyle 3025=55\times 55}$

${\displaystyle 3125={\dfrac {5\times 5^{5}}{5}}}$

${\displaystyle 161051=11^{5}=\left({\dfrac {55}{5}}\right)^{5}}$