Inverse Logarithmic Series

If $y=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots$, show that $x=y+\frac{y^2}{2!}+\frac{y^3}{3!}+\dots$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XVII. Exponential and Logarithmic Series, Examples XVII, No. 4