Choose the one alternative that best completes the statement or answers the question.
Solve the equation. Express irrational answers in exact form and as a decimal rounded to 3 decimal places.
-$$\left( \frac { 1 } { 2 } \right) ^ { x } = 7 ^ { 1 - x }$$
A) $\frac { \ln \left( \frac { 1 } { 2 } \right) + \ln 7 } { \ln 7 } \approx 0.644$
B) $\frac { \ln 7 } { \ln \left( \frac { 1 } { 2 } \right) + \ln 7 } \approx 1.553$
C) $\ln \left( \frac { 1 } { 2 } \right) - \ln 7 \approx - 2.639$
D) $\frac { \ln 14 } { \ln 28 } \approx 0.792$

Question

Quizplus · Accepted Answer

To solve the equation $\left( \frac { 1 } { 2 } \right) ^ { x } = 7 ^ { 1 - x }$, we can use logarithms. Taking the natural logarithm of both sides gives $x \ln \left( \frac { 1 } { 2 } \right) = (1 - x) \ln 7$. Solving for $x$ gives $x = \frac { \ln 7 } { \ln \left( \frac { 1 } { 2 } \right) + \ln 7 }$, which matches choice B.

Choose the One Alternative That Best Completes the Statement or Answers