(c) 3.
a > 0, b > 0, c > 0 ⇒ loga, logb, logc are all defined.
Also alogb – logc, blogc–loga, cloga–logb are all positive quantities.
∴ Applying AM > GM, we have
\(\frac13\)[ alogb – logc + blogc–loga + cloga–logb ] > [alogb – logc. blogc–loga. cloga–logb]\(\frac13\) ...(i)
Let x = alog b – log c . blog c – log a. clog a – log b
⇒ log x = (log b – log c) log a + (log c – log a) log b + (log a – log b) log c
Now, loge x = 0 ⇒ x = e0 = 1.
∴ (i) ⇒ \(\frac13\)[ alogb – logc + blogc–loga + cloga–logb ] > 1
⇒ alog b – log c + blog c – log a + clog a – log b > 3
⇒ The least value of alog b – log c + blog c – log a + clog a – log b is 3.