Answer
$\text{Exact: } \dfrac{ \ln 7}{\ln 0.6 + \ln 7 }$
$\text{Approximately: } 1.356$
Work Step by Step
Since $\dfrac{3}{5} = 0.6$, the equation can be written as:
$$(0.6)^x=7^{1-x}$$
Take the natural logarithm of both sides:
$\ln 0.6^{x} = \ln 7^{1-x}$
WIth $\ln M^r = r \ln M$, then:
$\therefore \ln 0.6^{x} = x \ln 0.6 \hspace{20pt} \text{and} \hspace{20pt} \ln 7^{1-x} = (1-x) \ln 7$
Thus, the equation above is equivalent to:
$x \ln 0.6 =(1-x) \ln 7$
$x \ln 0.6 = \ln 7 -x \ln 7$
Combine the $x$ terms:
$x \ln 0.6+ x \ln 7 = \ln 7$
Factor out $x$ (the common factor) then solve for $x$:
$x(\ln 0.6 + \ln 7 ) = \ln 7\\\\$
$x = \boxed{\dfrac{ \ln 7}{\ln 0.6 + \ln 7 } \approx 1.356}$
