Question

find the range of x.

Answer 1

\(\large \color{black}{\begin{align}& \{a,b\}\in \mathbb{R^{>0}},\ \ a>b,\ \ \normalsize \text{then }\ \ a^{1/x}>b^{1/x} \hspace{.33em}\\~\\
& \normalsize \text{find the range of }\ x .\hspace{.33em}\\~\\
\end{align}}\)

Answer 2

i think x>0

Answer 3

example http://www.wolframalpha.com/input/?i=solve+87%5E%281%2Fx%29%3E78%5E%281%2Fx%29

Answer 4

\[a^{1/x}\gt b^{1/x}\]

divide \(b^{1/x}\) both sides and get
\[\left(\frac{a}{b}\right)^{1/x}\gt 1\]

taking log both sides
\[x*\ln\left(\frac{a}{b}\right) \gt 0\]
since \(\ln\left(\frac{a}{b}\right) \gt 0\) for \(a\gt b\), dividing it both sides wont flip the signs :
\[x\gt 0\]

Answer 5

but after taking \(\ln \) on both sides it should be this ?

\(\dfrac1x \times \ln\left(\dfrac{a}{b}\right) \gt 0\)

Answer 6

Ahh sry, it was just a typo

Answer 7

fixed :

\[a^{1/x}\gt b^{1/x}\]

divide \(b^{1/x}\) both sides and get
\[\left(\frac{a}{b}\right)^{1/x}\gt 1\]

taking log both sides
\[\frac{1}{x}*\ln\left(\frac{a}{b}\right) \gt 0\]
since \(\ln\left(\frac{a}{b}\right) \gt 0\) for \(a\gt b\), dividing it both sides wont flip the signs :
\[\frac{1}{x}\gt 0\]
Multipilying \(x^2\) both sides gives
\[x\gt 0\]