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Art [367]
3 years ago
5

How do you solve x+4=10^x

Mathematics
1 answer:
raketka [301]3 years ago
3 0
Two ways to solve the problem:

A. By graphing f(x)=10^x and g(x)=x+4.
The intersection(s) will be the solution.
See graph below, approximate solutions: (-4,0),(0.66, 4.67)

B. Refine approximate solutions using Newton's method
Let h(x)=f(x)-g(x) = 10^x-x-4 ...............(1)
we calculate the derivative, h'(x) = log(10)*10^x-1   [ note:log(x) means ln(x) ]
and use Newton's iterative formula to find successive approximations to the root, basically refining the approximate solutions.
The iterative formula for nth approximation x_n is given by
x_n = x_{n-1} - h(x_n) / h'(x_n)....(2)

Using initial approximation (-4,0), we have x0=-4
x1
=x0-h(x0)/h'(x0)
=-4 - h(-4)/h'(-4)
=-4 - (10^(-4)-(-4)-4)/(log(10*10^(-4)-1)
=-4 - (1/10000)/(log(10)/10000-1)
=-3.9999000

Repeating the same for second approximation, x2, we get
x2=-3.99989997696619 which is accurate to 14 places after decimal

Now we can refine the other approximate solution, x0=0.66 to find
x1=0.669356
x2=0.6692468481102326
x3=0.669246832877748
x4=0.6692468328777476
So we will accept x=0.669246832877748

So the solutions are S={-3.99989997696619,0.6692468328777476}  both accurate to 14 places after the decimal

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