real analysis - To find the sum of the series $,1+ frac{1}{3cdot4}+frac{1}{5cdot4^2}+frac{1}{7cdot4^3}+ldots$

The answer given is $\log 3$.
Now looking at the series

$$\begin{align} 1+ \dfrac{1}{3\cdot4}+\dfrac{1}{5\cdot4^2}+\dfrac{1}{7\cdot4^3}+\ldots &= \sum\limits_{i=0}^\infty \dfrac{1}{\left(2n-1\right)\cdot4^n} \\ \log 3 &=\sum\limits_{i=1}^\infty \dfrac{\left(-1\right)^{n+1}\,2^n}{n} \end{align}$$

How do I relate these two series?

Answer

Hint: a common series that is used for computing log of any real number is
$$\log\left(\frac{1+x}{1-x}\right)=2\left(x+\frac{x^3}3+\frac{x^5}5+\frac{x^7}7+\dots\right)$$
$u=\frac{1+x}{1-x}\iff x=\frac{u-1}{u+1}$