Intersection Point of Two Logarithmic Graphs
Algebra · 25th PMO Qualifying Stage
Let $k>1$. The graphs of the functions$$f(x)=\log\left(\sqrt{x^2+k^3}+x\right)$$and$$g(x)=2\log\left(\sqrt{x^2+k^3}-x\right)$$have a unique point of intersection $(a,b)$. Find $2a$.
A$\sqrt{k^3-k+1}$
B$k^{3/2}-k^{1/2}+1$
C$k^2+k+1$
D$k^2-k$