The power of triples #1

The following is a proof that does not use logarithms or calculators. We know that $8991>64.$ Then $3^{4} 111>4^{3}.$ Multiplying both sides by $111^{3}$ $3^{4}*111^4>4^{3}*111^3$ Then $333^{4}>444^{3}.$ Now, raising both sides to 111 we get that $333^{444}>444^{333}.$

Let's assume to begin with that they're equal. It follows that we have the following: ${ 333 }^{ 444 }=444^{ 333 }$ We can simplify this further using log rules:

${ 333 }^{ 444 }=444^{ 333 }\\ \log _{ 333 }{ { (333 }^{ 444 }) } =\log _{ 333 }{ (444^{ 333 }) } \\ 444=333\log _{ 333 }{ (444) } \\ \frac { 444 }{ 333 } =\log _{ 333 }{ (333\times \frac { 4 }{ 3 } ) } \\ \frac { 4 }{ 3 } =\log _{ 333 }{ (333) } +\log _{ 333 }{ (\frac { 4 }{ 3 } ) } \\ \frac { 4 }{ 3 } =1+\log _{ 333 }{ (\frac { 4 }{ 3 } ) } \\ \frac { 1 }{ 3 } =\log _{ 333 }{ (\frac { 4 }{ 3 } ) } \\ { 333 }^{ \frac { 1 }{ 3 } }=\frac { 4 }{ 3 } \\ \sqrt [ 3 ]{ 333 } =\frac { 4 }{ 3 }$

As you can see, we have the cube root of 333 = 4/3, which is obviously not true as 333 lies between $6^3$ and $7^3$ . As 6 and 7 are much greater than 4/3, the LHS must be greater, i.e. $333^{444}$ is much greater than $444^{333}$ .