If ax = (a/k)y = km then 1/2 - 1/y =A) 0 B) 1C) m D) 1/m

1.	If ax = (a/k)y = km then 1/2 - 1/y =A) 0 B) 1C) m D) 1/m
Answer» Correct option is (D) 1/m \(a^x=(\frac {a}{k})^y=k^m\) \(\Rightarrow\) \(log\,(a^x)=log\,(\frac {a}{k})^y=log\,(k^m)\) (By taking log) \(\Rightarrow\) x log a = y log \(\frac ak\) = m log k \(\Rightarrow\) x log a = m log k and y log \(\frac ak\) = m log k \(\Rightarrow\) \(x=\frac{m\;log\,k}{log\,a}\) \(and\,y=\frac{m\;log\,k}{log\,\frac ak}\) \(=\frac{m\;log\,k}{log\,a-log\,k}\) \((\because\) \(log\,\frac AB\) = log A - log B) Now, \(\frac{1}{x}-\frac{1}{y}\) \(=\frac{log\,a}{m\;log\,k}\) \(-\frac{log\,a-log\,k}{m\;log\,k}\) \(=\frac{log\,a-log\,a+log\,k}{m\;log\,k}\) \(=\frac{log\,k}{m\;log\,k}\) \(=\frac{1}{m}\) Correct option is D) 1/m

Answer»

Correct option is (D) 1/m

\(a^x=(\frac {a}{k})^y=k^m\)

\(\Rightarrow\) \(log\,(a^x)=log\,(\frac {a}{k})^y=log\,(k^m)\) (By taking log)

\(\Rightarrow\) x log a = y log \(\frac ak\) = m log k

\(\Rightarrow\) x log a = m log k and y log \(\frac ak\) = m log k

\(\Rightarrow\) \(x=\frac{m\;log\,k}{log\,a}\) \(and\,y=\frac{m\;log\,k}{log\,\frac ak}\) \(=\frac{m\;log\,k}{log\,a-log\,k}\)

\((\because\) \(log\,\frac AB\) = log A - log B)

Now, \(\frac{1}{x}-\frac{1}{y}\) \(=\frac{log\,a}{m\;log\,k}\) \(-\frac{log\,a-log\,k}{m\;log\,k}\)

\(=\frac{log\,a-log\,a+log\,k}{m\;log\,k}\)

\(=\frac{log\,k}{m\;log\,k}\) \(=\frac{1}{m}\)

Correct option is D) 1/m

Discussion