I was solving the definite integral $\int_{\sqrt7}^{2\sqrt7}{1\over\sqrt {x^2-7}}dx$, and came out with the intermediate step $\int_{\sqrt7}^{2\sqrt7}\sec\theta\ d\theta$, which led me to finish off with $$\ln \left|\sec\theta +\tan\theta \right|\bigg|_{arc\sec(1)}^{arc\sec(2)}$$
$$=\ln\left|x+\sqrt{x^2-7}\right|\bigg|_{\sqrt7}^{2\sqrt7}-\require{cancel} \cancel{\ln\sqrt7}$$
$$=\ln\left|{2\sqrt7+\sqrt{21}}\over\sqrt7\right|$$
$$=\ln\left|{2+\sqrt{3}}\right|$$
However, my book, after the step of $\int_{\sqrt7}^{2\sqrt7}\sec\theta\ d\theta$, seems to go in a completely different direction and comes out with $$={1\over2}\ln\left|{7+4\sqrt{3}}\right|$$
Upon checking them, I see that they are equal, and my question is, is there any property of logarithms or algebraic reason with which I can recognize them as equal, without solving the integral $\int_{\sqrt7}^{2\sqrt7}\sec\theta\ d\theta$ in a different way?