Question

Using the change of base formula, the following expression was created: log2128log232
. Identify the base of the original expression before the change of base formula was used.(1 point)
The base of the original expression was
.

Answer 1

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:

1. **Given Expression**: \(\log_2\) indicates a logarithm with base 2.
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.**