Wouldn't it be easy to find private keys with the Diffie Hellman exchange?

Question

I recently watched this video by Computerphile, where Mike explains the mathematics of the Diffie Hellman exchange. I've been wondering, since as explained $g$, $g^a$ and $g^b$ is public, can't you just do this equation $$ \frac{1}{log_{(g^a)}(g)} = a $$ and $$ \frac{1}{log_{(g^b)}(g)} = b $$ to find their private keys? I might be dumb, please explain this.

Answer 1

Computing the discrete log is the hard part. Remember that the logarithms are taking place in the group, not the real numbers.