Can anybody tell me how to find the value of $\alpha$?

Question

I don't know how to find the value of $\alpha$: $$(\sqrt{3}+\sqrt{2})^\alpha + (\sqrt{3}+\sqrt{2})^\alpha = 10$$ I tried to simplify it to $$(\sqrt{3}+\sqrt{2})^\alpha=5$$ then take the ($\ln$) of both sides $$\ln{(\sqrt{3}+\sqrt{2})^\alpha} = \ln{5}$$ then $$\alpha\ln{(\sqrt{3}+\sqrt{2})} = \ln{5}$$ then $$\alpha = \frac{\ln{5}}{\ln{(\sqrt{3}+\sqrt{2})}}$$ is that correct?