To determine the base of the original logarithmic expression before the change of base formula was used, we can analyze the given expression:
\[\log_2 128 \cdot \log_2 32.\]
The change of base formula for logarithms states that:
\[ \log_b a = \frac{\log_k a}{\log_k b} \]
for any positive base \( k \). When applying this to \(\log_2 128\) and \(\log_2 32\), it suggests that they were transformed into a form involving a different base (let's denote it as \( k \)).
Thus, if we denote the original logarithm expression as \( \log_b a \), it implies:
- Given Expression: \(\log_2\) indicates a logarithm with base 2.
- Identifying Original Base: Since the terms \(\log_2 128\) and \(\log_2 32\) can also be represented using any logarithmic base \( k \), and if we denote the original base \( b \), we can look for a common relationship between them.
To explicitly identify the original base:
The expression simplifies to \( \log_2(128) = 7 \) and \( \log_2(32) = 5 \). If both were rewritten using another base \( b \):
\[ \log_b 128 = \frac{\log_k 128}{\log_k b} \quad \text{and} \quad \log_b 32 = \frac{\log_k 32}{\log_k b}. \]
However, since we are only examining the product \( \log_2 128 \cdot \log_2 32 \), which still essentially keeps the base as 2. Thus, it's evident that:
The original base of the expression before the change of base formula was:
2.