The sum of the given series is?

Question

$\log_42-\log_82+\log_{16}2-\log_{32}2+...$
In the given solution, the answer is given as $1-\ln2$. How do I arrive at this solution? Thank you!

Answer 1

$\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad $, see series expansion

Put $x = 1 $ and you get the required form