$$f(x)=\log(x+ \sqrt{x^{2} -1})=\log\big((x- \sqrt{x^{2} -1})^{-1}\big)=-\log(x- \sqrt{x^{2} -1})$$
حال اگر با $3f(x)$ شروع کنیم:
$$\color{blue}{ 3f(x)=-3\log(x- \sqrt{x^{2} -1})=-\log(x- \sqrt{x^{2} -1})^{3}}$$
$$=-\log(x^3-3x^2\sqrt{x^{2} -1}+3x(x^2-1)-(x^2-1)\sqrt{x^{2} -1})$$
$$=\color{green}{-\log(4x^3-3x-(4x^2-1)\sqrt{x^{2} -1})=-\log(4x^3-3x-\sqrt{(1-4x^2)(x^{2} -1)})}$$
$$=-\log(4x^3-3x-\sqrt{16x^6-24x^4+9x^2 -1})$$
$$\color{red}{=-\log(4x^3-3x-\sqrt{ (4x^3-3x)^{2} -1})=\log(4x^3-3x+\sqrt{ (4x^3-3x)^{2} -1})=f(4x^3-3x)}$$
