CAT 2024 Slot 3 QA Question 17: If $3^{a} = 4,\;4^{b} = 5,\;5^{c} = 6,\;6^{d} = 7,\;7^{e} = 8$ and $8^{f} = 9$, then the value of the product $abcdef$ is

Question

If $3^{a} = 4,\;4^{b} = 5,\;5^{c} = 6,\;6^{d} = 7,\;7^{e} = 8$ and $8^{f} = 9$, then the value of the product $abcdef$ is

Accepted Answer

Rewrite each exponent as a logarithm.

$a = \log_{3}4$
$b = \log_{4}5$
$c = \log_{5}6$
$d = \log_{6}7$
$e = \log_{7}8$
$f = \log_{8}9$

The product telescopes because $\log_m n	imes \log_n p = \log_m p$.

$abcdef = \log_3 4	imes \log_4 5	imes \log_5 6	imes \log_6 7	imes \log_7 8	imes \log_8 9$

If 3a=4, 4b=5, 5c=6, 6d=7, 7e=83^{a} = 4,\;4^{b} = 5,\;5^{c} = 6,\;6^{d} = 7,\;7^{e} = 83a=4,4b=5,5c=6,6d=7,7e=8 and 8f=98^{f} = 98f=9, then the value of the product abcdefabcdefabcdef is