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The functional equation $f(y-x)-g\left(xy\right)=h\left(1/x-1/y\right)$ is solved for general solution. The result is then applied to show that the three functional equations $f\left(xy\right)=f\left(x\right)+f\left(y\right)$, $f(y-x)-f\left(xy\right)=f(1/x-1/y)$ and $f(y-x)-f\left(x\right)-f\left(y\right)=f(1/x-1/y)$ are equivalent. Finally, twice differentiable solution functions of the functional equation $f(y-x)-g_1\left(x\right)-g_2\left(y\right)=h\left(1/x-1/y\right)$ are determined.