how to solve this equation using logarithm, if not possible how to solve it?

Question

how to solve the following equation:

$$0.2948(1-(1+x)^{-5})=x$$

I know to satisfy this equation $x$ should be equal to 0.145 but how i can get there please help!

Answer 1

HINT: expanding all and factorizing we get $$-x \left(2500 x^5+11763 x^4+21315 x^3+17630 x^2+5130 x-1185\right)=0$$ this is an polynomial in degree six. One solution is $x=0$ for the other you will need a numerical method.

Answer 2

Newton's method is very efficient for this sort of equation, especially when you have a good initial estimate. It's easy to guess $x\approx 0.2$ to begin. So, using Newton's iteration $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)},$$with $$f(x)=0.2948\left(1-\frac1{(1+x)^5}\right)-x,\quad\quad f'(x)=\frac{1.474}{(1+x)^6}-1,$$and $x_0=0.2$, we can successively compute $x_1,x_2,$ and so on. Just two iterations will be enough to give good accuracy, and three is plenty.