Joy used the change of base formula to rewrite

Question

log
5
20
as
log
20
log
5
. Payne rewrote it as
ln
20
ln
5
. Who is correct? Payne, Joy, both, neither? Explain how you know.

Answer 1

Let's clarify the change of base formula for logarithms. The change of base formula states that:

\[ \log_b a = \frac{\log_k a}{\log_k b} \]

for any positive base \( k \) different from 1.

Now, Joy used the formula to rewrite \(\log_5 20\) as:

\[ \log_5 20 = \frac{\log 20}{\log 5} \]

Here, Joy uses a base of 10 (common logarithm), which is valid.

Payne rewrote it as:

\[ \log_5 20 = \frac{\ln 20}{\ln 5} \]

Here, Payne uses the natural logarithm (base \( e \)), which is also valid.

Both expressions are correct because the change of base formula allows you to use any logarithmic base. Consequently, both Joy and Payne are correct as they simply expressed the same logarithmic relationship using different bases. So, the answer is:

Both are correct.