= Solution
The <Gelfond–Schneider theorem> says that if $a\ne0,1$ is algebraic and $b$ is algebraic irrational, every number $e^{b\lambda}$ with $e^\lambda=a$ is transcendental. Thus the statement includes every determination of the power, not merely a chosen principal branch.
Here is an auxiliary-function outline with the arithmetic and analytic steps specified. Assume $v=e^{b\lambda}$ were algebraic, and put $K=\mathbb Q(a,b,v)$, $d=[K:\mathbb Q]$. For a large <integer> $T$, choose degrees $L_1\asymp T\log T$ and $L_2\asymp T/\log T$, with constants large enough that $(L_1+1)(L_2+1)>2d(T+1)^2$. Apply <Siegel lemma> to impose
$$
P(s+tb,a^sv^t)=0\qquad(0\leq s,t\leq T)
$$
on a nonzero <polynomial> $P(X,Y)\in\mathbb Z[X,Y]$ of these two degrees. Each field equation gives at most $d$ rational equations. Clearing denominators and bounding conjugates gives equation-coefficient logarithmic size $O(L_1\log T+L_2T)$; the surplus of variables therefore permits coefficient height $H$ with
$$
\log H=O(L_1\log T+L_2T)=o(T^2).
$$
The <entire function> $F(z)=P(z,e^{\lambda z})$ is nonzero: its <polynomial>-exponential terms have distinct exponents $j\lambda$, and their independence follows by the differential-operator argument in question 2. Here $\lambda\ne0$. Its maximum modulus satisfies $\log M_F(R)=O(\log H+L_1\log R+L_2R)$. The grid points $s+tb$ are distinct because $b$ is irrational, and lie in a disk of radius $(1+|b|)T$.
Suppose the vanishing grid has been extended to $0\leq s,t\leq B$, for some $B\geq T$. Divide $F$ by the product of its $(B+1)^2$ known linear zero factors and use the maximum principle on a disk of radius $C B$, with $C$ large and fixed. At a point of the doubled grid, the quotient estimate gives
$$
\log|F(s+tb)|\leq-c_1B^2+O(\log H+L_1\log B+L_2B),\qquad0\leq s,t\leq2B,
$$
if the value is nonzero. But this value equals the <algebraic number> $P(s+tb,a^sv^t)$. Clearing its denominators and taking its norm gives the opposite bound
$$
\log|F(s+tb)|\geq-C_1(\log H+L_1\log B+L_2B).
$$
For sufficiently large initial $T$, the terms on the right are $o(B^2)$ uniformly for every $B\geq T$: $L_2/B=O(1/\log T)$ and $L_1\log B/B^2=O((\log T)^2/T)$. Thus every doubled-grid value must be zero. Iterating doubles the grid indefinitely. This supplies quadratically many distinct zeros in disks of linearly growing radius, whereas <Jensen's formula> and the exponential-<polynomial> growth bound allow only $O(R)$ zeros for a nonzero $F$. The contradiction proves the theorem. This is the <Schneider lattice proof of the Gelfond–Schneider theorem>.
Now fix any <logarithms> of algebraic $\alpha,\beta\ne0,1$. Their values are nonzero. If $q=\log\alpha/\log\beta$ were algebraic irrational, then $e^{q\log\beta}=\alpha$ would be algebraic, contradicting the theorem with base $\beta$. Hence
$$
\boxed{\frac{\log\alpha}{\log\beta}\text{ is either rational or transcendental}.}
$$
For arbitrarily many <logarithms> the qualitative <Baker theorem on algebraic linear combinations of logarithms> states that <logarithms> of <algebraic numbers> which are linearly independent over $\mathbb Q$ remain independent over $\overline{\mathbb Q}$, even after adjoining one to the list. Equivalently a nonzero homogeneous algebraic-coefficient linear combination of such <logarithms> is transcendental. Rational relations are therefore the only source of homogeneous algebraic relations among the chosen <logarithms>.
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