If there are integers m and n such that

æ
ç
è

ö
÷
ø

æ
ç
è

ö
÷
ø

then

log2

log5/4

is rational and 5^m=2^n+2m but this is impossible as 2 and 5 are prime so there cannot be integer solutions of this equation.

Now suppose

æ
ç
è a
b
ö
÷
ø m

= æ
ç
è c
d
ö
÷
ø n

then a^mdⁿ=b^mcⁿ and, as a and b are coprime, a^m divides cⁿ and cⁿ divides a^m so a^m=cⁿ. Similarly b^m=dⁿ.

This implies that a^m and cⁿ have the same prime factors. Write a = p₁^u₁...p_k^u_k and c = p₁^v₁...p_k^v_k and for all j we have mu_j = nv_j so that

u₁
v₁
= u₂
v₂
= ... = u_k
v_k
.

Similarly for b and d.

This is a very special relationship between a and c and also between b and d so in general this does not happen and there are no positive integers m,n such that

æ
ç
è a
b
ö
÷
ø m

= æ
ç
è c
d
ö
÷
ø n

.

Conversely suppose a = p₁^u₁...p_k^u_k and c = p₁^v₁...p_k^v_k and

u₁
v₁
= u₂
v₂
= ... = u_k
v_k
.

We call this common ratio ⁿ/_m then u_jm=v_jn for all j and a^m = bⁿ.

Similarly if corresponding ratios of the powers of the prime factors of b and d are constant and also equal to ⁿ/_m then b^m=dⁿ giving

æ
ç
è a
b
ö
÷
ø m

= æ
ç
è c
d
ö
÷
ø n

.