Mathematica can't simplify some logarithmic expressions
Use RootApproximant
y1 = Log[Sqrt[2] + 1, 5 Sqrt[2] + 7];
y1 // RootApproximant
(* 3 *)
Verifying,
y1 == 3 // FullSimplify
(* True *)
y2 = Log[2 Sqrt[2] + 3, 17 - 12 Sqrt[2]];
y2 // RootApproximant
(* -2 *)
Verifying,
y2 == -2 // FullSimplify
(* True *)
You can use Reduce
to solve the same problem in equation form:
Reduce[(Sqrt[2] + 1)^n == 5 Sqrt[2] + 7, n, Integers]
n == 3
Reduce[(Sqrt[2] + 1)^n == 17 - 12 Sqrt[2], n, Integers]
n == -4
Reduce[
(5 Sqrt[2] + 7)^(c*n) == (3 + 2 Sqrt[2])^a (17 - 12 Sqrt[2])^b
\[And] a != 0 \[And] b != 0 \[And] c != 0,
n, Reals
]
a != 0 && b != 0 && c != 0 && n == -((-2 a + 4 b)/(3 c))