How do you solve 2log(2x) = 1 + loga?

Aug 15, 2016

$x = \frac{\sqrt{10 a}}{2}$ (Assuming $\log = {\log}_{10}$)

Explanation:

$2 {\log}_{10} \left(2 x\right) = 1 + {\log}_{10} a$

$2 {\log}_{10} 2 x - {\log}_{10} a = 1$

$2 {\log}_{10} 2 x - 2 {\log}_{10} {a}^{\frac{1}{2}} = 1$

$2 {\log}_{10} \left(\frac{2 x}{\sqrt{a}}\right) = 1$

${\log}_{10} \left(\frac{2 x}{\sqrt{a}}\right) = \frac{1}{2}$

$\frac{2 x}{\sqrt{a}} = {10}^{\frac{1}{2}}$

$2 x = \sqrt{10 a}$

$x = \frac{\sqrt{10 a}}{2}$