Five Fives/Table of 5s

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Zero 5s

$1=\ln {e}$

$9+4{\sqrt {5}}=\phi \left(\phi ^{5}\right)$

One 5

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

${\sqrt {5}}$

$\phi ^{n}{\sqrt {5}}=\phi ^{n+1}+\phi ^{n-1}$

$-1=\cos {\left(5\pi \right)}$

$0=\sin {\left(5\pi \right)}$

$1=x^{\sin {\left(5\pi \right)}}$

$2=\phi \times \phi \times \phi -{\sqrt {5}}$

$5$

$15=5!!$

$120=5!$

Two 5s

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

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

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

$3=\phi ^{5}-5\phi$

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

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

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

$10=5+5$

$11=\phi ^{5}-\phi ^{-5}$

$16={\dfrac {\ln {5}}{\ln {\left({\sqrt {\sqrt {\sqrt {\sqrt {5}}}}}\right)}}}$

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

$25=5\times 5$

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

$55$

$115=5!-5$

$125=5!+5$

$600=5\times 5!$

$3125=5^{5}$

$11981655542024930675232002=\phi ^{5!}-\phi ^{-5!}$

$12696403353658275925965100847566516959580321051449436762275840000000000000=55!$

Three 5s

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

${\dfrac {3}{5}}={\dfrac {1}{5}}\left(\phi ^{5}-5\phi \right)$

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

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

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

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

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

$5=5-5+5$

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

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

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

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

$7=5+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}$

$8=\phi ^{5}-5\phi +5$

$9=5+{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {\sqrt {5}}}\right)}}}$

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

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

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

$13=5!!-{\dfrac {\ln {\left(5\right)}}{\ln {\left({\sqrt {5}}\right)}}}$

$15=5+5+5$

$16=\phi ^{5}-\phi ^{-5}+5$

$18=5!!+\phi ^{5}-5\phi$

$19={\dfrac {5!}{5}}-5$

$20=5\times 5-5$

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

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

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

$25={\sqrt {5}}\left(\phi ^{5}+\phi ^{-5}\right)$

$25=5+5+5!!$

$26=\left(\phi ^{5}-\phi {-5}\right)+5!!$

$29={\dfrac {5!}{5}}+5$

$30=5\times 5+5$

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

$50=55-5$

$50=5(5+5)$

$55=5\left(\phi ^{5}-\phi ^{-5}\right)$

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

$95=5!-5\times 5$

$96=5!-{\dfrac {5!}{5}}$

$110=5!-5-5$

$120=5!+5-5$

$123=\phi ^{5}-5\phi +5!$

$130=5!+5+5$

$145=5!+5\times 5$

$243=\left(\phi ^{5}-5\phi \right)^{5}$

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

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

$3005=5^{5}-5!$

$9765625=\left(5\times 5\right)^{5}$

$503284375=55^{5}$

$298023223876953125=5^{5\times 5}$

Four 5s

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

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

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

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

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

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

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

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

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

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

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

$20=5+5+5+5$

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

$34=\phi ^{5+5}-55\phi$

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

$35=5\times 5+5+5$

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

$50=5\times 5+5\times 5$

$52=55-\phi ^{5}+5\phi$

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

$70=5!+5-55$

$89={\dfrac {\phi \left(\phi ^{5+5}\right)-55}{\phi }}$

$120=5\times 5\times 5-5$

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

$125=5{\sqrt {5}}\left(\phi ^{5}+\phi {-5}\right)$

$130=5\times 5\times 5+5$

$140=5!+(5\times 5)-5$

$150=5\left(5\times 5+5\right)$

$160=5\left({\dfrac {\ln {5}}{\ln {\sqrt {5}}}}\right)^{5}$

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

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

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

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

$605=55\left(\phi ^{5}-\phi ^{-5}\right)$

$3000=\left(5^{5}-5\right)-5!$

$3025=55\times 55$

$3070=5^{5}-55$

$3100=5^{5}-5\times 5$

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

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