Answer
$p \approx -0.349$
Work Step by Step
$\log (2p+1) = -0.5$
$10^{-0.5} = 2p + 1$
$10^{-0.5} - 1 = 2p$
$p = \frac{10^{-0.5}-1}{2}$
$p = -0.341886...$
$p \approx -0.349$
Check:
$\log_{10} (2(\frac{10^{-0.5}-1}{2})+1) \overset{?}{=} -0.5$
$\log_{10} (10^{-0.5}-1+1) \overset{?}{=} -0.5$
$\log_{10} (10^{-0.5}) \overset{?}{=} -0.5$
$-0.5\log_{10} (10) \overset{?}{=} -0.5$
$-0.5(1) \overset{?}{=} -0.5$
$-0.5 = -0.5$
