Linear forms in logarithms and applications to Diophantine problems

The theory of linear forms in logarithms (Baker's method) is a very useful tool for solving Diophantine equations. After introducing some background and important milestones, I will present recent results on linear forms in two $p$-adic logarithms, where an upper bound for the $p$-adic valuation of two integral powers of algebraic numbers (that is $v_p(\alpha_1^{b_1} - \alpha_2^{b_2})$) is established. The bound has a good dependence on the logarithm of $b_1$ and $b_2$. Finally, I will briefly present the solutions to a variant of Pillai's problem. (The last part is a joint work with I. Pink and V. Ziegler.)