How do you solve 3^x + 4 = 7^x - 1 3x+4=7x1?

1 Answer
Oct 8, 2016

x approx 1.08777x1.08777

Explanation:

Calling

f(x) = 7^x-3^x-5f(x)=7x3x5 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(xx0)

or

f(x_1) approx f(x_0)+((df)/(dx))_(x_0)(x_1-x_0)f(x1)f(x0)+(dfdx)x0(x1x0)

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=xkf(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