CAT 2024

Slot 3

QA

Question & Solution

Taking a log for each of the expressions, we get the following:

$\log_34=a,\ \log_45=b,\ \log_56=c,\ \log_67=d,\ \log_78=e,\ \log_89=f$

The expression $abcef$ would then be: $\log_34\times\ \log_45\times\ \log_56\times\ \log_67\times\ \log_78\times\ \log_89$

Next, we can use this property of log: $\frac{\log_ba}{\log_bc}=\log_ca$
Using this, we get:

$\frac{\log\ 4}{\log\ 3}\times\ \frac{\log\ 5}{\log\ 4}\times\ \frac{\log\ 6}{\log\ 5}\times\ \frac{\log\ 7}{\log\ 6}\times\ \frac{\log\ 8}{\log\ 7}\times\ \frac{\log\ 9}{\log\ 8}$

All the terms will cancel out except: $\frac{\log\ 9}{\log\ 3}=\log_39=2$

Therefore, 2 is the correct answer.