© 2019 Elsevier Inc. In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity ht=∇⋅([Formula presented]∇e [Formula presented])=∇⋅([Formula presented]∇e−∇⋅([Formula presented])) where total energy E=∫|∇h| is the total variation of h. Using a logarithmic correction for total energy E=∫|∇h|ln⁡|∇h|dx and gradient flow structure with a suitable defined functional, we prove the one dimensional evolution variational inequality solution preserves a positive gradient hx which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity hxx+ to happen.