Calling
f(x) = 7^x-3^x-5f(x)=7x−3x−5 Expanding in Taylor series around x_0x0
f(x) approx f(x_0)+((df)/(dx))_(x_0)(x-x_0)f(x)≈f(x0)+(dfdx)x0(x−x0)
or
f(x_1) approx f(x_0)+((df)/(dx))_(x_0)(x_1-x_0)f(x1)≈f(x0)+(dfdx)x0(x1−x0)
If x_0x0 is near a function zero, then f(x_1) approx 0f(x1)≈0 so
x_(k+1)=x_k-f(x_k)/((df)/(dx))_(x_k)xk+1=xk−f(xk)(dfdx)xk
Here
((df)/(dx))_(x_k)=7^(x_k) Log(7)-3^(x_k) Log(3) (dfdx)xk=7xklog(7)−3xklog(3)
Begining with x_0 = 1x0=1 we have
x_0=1x0=1
x_1=1.09685x1=1.09685
x_2=1.08786x2=1.08786
x_3=1.08777x3=1.08777
x_4=1.08777x4=1.08777