## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$\text{Exact: } \dfrac{ \ln 5 - \ln 2}{\ln 2 + 2 \ln 5}$ $\text{Approximately: } 0.234$
Take the natural logarithm of both sides: $\ln 2^{x+1} = \ln 5^{1-2x}$ Since $\ln M^r = r \ln M$, then: $\ln 2^{x+1} = (x+1) \ln 2 \hspace{20pt} \text{and} \hspace{20pt} \ln 5^{1-2x} = (1-2x) \ln 5$ Thus the equation above is equivalent to $(x+1) \ln 2 =(1-2x) \ln 5$ Distribute to obtain: $x \ln 2 + \ln 2 = \ln 5 -2x \ln 5$ Combine the $x$ terms: $x \ln 2+ 2x \ln 5 = \ln 5 - \ln 2$ Factor out the $x$ (the common factor) then solve for $x$: $x(\ln 2 + 2 \ln 5 ) = \ln 5 - \ln 2\\$ $x = \boxed{\dfrac{ \ln 5 - \ln 2}{\ln 2 + 2 \ln 5} \approx 0.234}$