(2x)-1/2ln(7x+15)=0

Question

find the value of x

Answer 1

Let f(x)=4x-ln(7x+15)=0, then we can use Newton's Method to solve this iteratively:

x_n+1=x_n-f(x_n)/f'(x_n) where xn is the nth iteration of x, and f'(x)=df/dx.

f'(x)=df/dx=4-7/(7x+15)

We need a starting value for x (x₀), so let x₀=0.

f(0)=-ln(15); f'(0)=4-7/15=53/15.

x₁=ln(15)/(4-7/15)=15ln(15)/53=0.766429 approx.

x₂=0.752235,

x₃=0.752231,

x₄=0.752231. To 6 decimal places x=0.752231.