Question

log base 12 of 18 equal to a , then find the value of log base 24 of 16

Answer 1

given, log_18 (12) = a ….(1)

We have to find the value of,

= log_24 (16)

= log_18 (16/) / log_18 (24)

= log_18 (2^4) / log_18 (8×3)

= 4log_18 (2) / {log_18 (8) + log_18 (3)}

= 4log_18 (2) / {log_18 (2^3) + log_18 (3)}

= 4log_18 (2) / {3log_18 (2) + log_18 (3)}

= 4/{3+log_18 (3)/log_18 (2)}

= 4/{3+log_2 (3)}

= 4/(3+p) ; where, p = log_2 (3) ….(2)

Now, from (1), we have,

log_2 (12) / log_2 (18) = a

=> log_2 (3×4) / log_2 (2×9) = a

=> log_2 (3×2^2) / log_2 (2×3^2) = a

=> {log_2 (3) + 2log_2 (2)} / {log_2 (2) + 2log_2 (3)} = a

=> (p+2×1)/(1+2p) = a

=> p+2 = a+2ap

=> p = (a-2)/(1–2a)

Therefore, from (2), we get,

log_18 (16) / log_18 (24) = 4/{3+(a-2)/(1–2a)}

= 4(1–2a)/(1–5a) . (Ans)


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