If \(\cfrac{log\,x}{a}=\cfrac{log\,y}{2}=\cfrac{log\,z}{5}=k\,\) and

1.	If \(\cfrac{log\,x}{a}=\cfrac{log\,y}{2}=\cfrac{log\,z}{5}=k\,\) and x4 y3 z-2 = 1, find ‘a’.log x /a = logy /2 = log z/5 = k
Answer» Let \(\cfrac{log\,x}{a}=\cfrac{log\,y}{2}=\cfrac{log\,z}{5}=k\,\) \(\therefore\) log x = ak, log y = 2k, log z = 5k ….(i) But x⁴ y³ z^-2 = 1 Taking log on both sides, we get log (x⁴ .y³ .z^-2 ) = log 1 \(\therefore\) log x⁴ + log y³ + log z^-2 = 0 \(\therefore\) 4 log x + 3 log y -2 log z = 0 \(\therefore\) 4(ak) + 3(2k) - 2(5k) = 0 ….[From(i)] \(\therefore\) 4ak + 6k - 10k = 0 \(\therefore\) 4ak = 4k \(\therefore\) a = 1

If \(\cfrac{log\,x}{a}=\cfrac{log\,y}{2}=\cfrac{log\,z}{5}=k\,\) and x4 y3 z-2 = 1, find ‘a’.log x /a = logy /2 = log z/5 = k

Answer»

Let \(\cfrac{log\,x}{a}=\cfrac{log\,y}{2}=\cfrac{log\,z}{5}=k\,\)

\(\therefore\) log x = ak, log y = 2k, log z = 5k ….(i)

But x⁴ y³ z^-2 = 1

Taking log on both sides, we get

log (x⁴ .y³ .z^-2 ) = log 1

\(\therefore\) log x⁴ + log y³ + log z^-2 = 0

\(\therefore\) 4 log x + 3 log y -2 log z = 0

\(\therefore\) 4(ak) + 3(2k) - 2(5k) = 0 ….[From(i)]

\(\therefore\) 4ak + 6k - 10k = 0

\(\therefore\) 4ak = 4k

\(\therefore\) a = 1