Logarithmic Relation Proof

Show that ${\log }_e(x+2h)=2{\log }_e(x+h)-{\log }_{e}x-\left\{\frac{h^2}{(x+h{)}^2}+\frac{h^4}{2(x+h{)}^4}+\frac{h^6}{3(x+h{)}^6}+\dots \right\}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XVII. Exponential and Logarithmic Series, Examples XVII, No. 16