143rd Problem 2017

Solution 1:

$(\dfrac{1}{2})^x=4$

$(\dfrac{1}{2})^x=(\dfrac{1}{2})^{-2}$

$\color{#D61F06}\boxed{x=-2}$ $\boxed{\text{answer}}$

Solution 2.

$(\dfrac{1}{2})^x=4$

taking the natural logarithm of both sides, we have

$\ln~(\dfrac{1}{2})^x=\ln~4$

$x\ln\dfrac{1}{2}=\ln~4$

$x=\dfrac{\ln~4}{\ln~0.5}$

$\color{#D61F06}\boxed{x=-2}$ $\boxed{\text{answer}}$

$\displaystyle \bigg(\frac 12\bigg)^x = 4$

$\displaystyle 2^{-x} = 2^2$

$\displaystyle - x = 2 \Longrightarrow x = - 2$

$x = \log_{\frac{1}{2}}(4)=-2$