I need to solve logarithmic equation that contains a variable inside and outside the logarithm. I want to solve the following equation for x:
$$x= \frac{(\log_2(1+2x)) \cdot (1+2x)}{4}-5 $$
Note1: The base of logarithm is $2$.
Note2: The equation is supposed to be closed form.
Note3: The constant values are not important and can be changed.
Best Answer
$$ \underbrace{\log(2x+1)}_{g(x)} = \underbrace{4x+20\over 2x+1}_{f(x)}$$
Since $f$ is decreasing for $x>-{1\over 2}$ and $g$ is increasing the equation has at most one solution.
Draw both graphs and you will find an aproximate solution.