TITLE: Cardinality of a generating set of a group?
QUESTION [6 upvotes]: In a paper that I am reading, the author claims that 

Any group $G$ has a generating set of cardinality $\log|\,G\,|$ or less.

Can someone show the ideas that went into this statement? 
The paper, although irrelevant to this particular question, can be found here and the statement is the first sentence of section 1.2 on the 4th page of the PDF.

REPLY [5 votes]: Hint: Let $|G|=n$. Take $e \neq g_1 \in G$ and consider $\langle g_1 \rangle$. Since $g_1 \neq e$, then $|\langle g_1 \rangle| \geq 2$. If $G = \langle g_1 \rangle$ then $G$ is generated by 1-element set, else let $g_2 \in G - \langle g_1 \rangle$. Consider $\langle g_1,g_2 \rangle$. Since $\langle g_1 \rangle$ is a proper subgroup of $\langle g_1,g_2 \rangle$ (it should have index at least 2), we obtain $|\langle g_1,g_2 \rangle| \geq 2\cdot2$. What we'll obtain after $\log n$ steps?