Master theorem proof about relationships $T(n)=aT(n/b)+f(n)$

Question

I am reading the proof of Master theorem from Cornell University lectures: https://www.cs.cornell.edu/courses/cs3110/2012sp/lectures/lec20-master/mm-proof.pdf

I am having problem in this step:

$$n^{log_ba-\epsilon} \frac{b^{\epsilon (log_bn+1)} - 1}{b^\epsilon - 1} = n^{log_ba-\epsilon} \frac{b^{\epsilon} n^\epsilon - 1}{b^\epsilon - 1}$$

So how did author claim:

$$b^{(log_bn+1)} = n^\epsilon$$

I understand it is some epsilon > 0, but how can you put an equal sign between two expressions like that?

Many thanks in advance.

Answer 1

Like this $$ \begin{align} b^{\epsilon(\log_b n + 1)} &= b^{\epsilon \log_b n} b^{\epsilon} \\ &= b^{\log_b n^{\epsilon}} b^{\epsilon} \\ &= n^{\epsilon}b^{\epsilon} \end{align} $$