How do you solve 1+4x=3^X ?

Question

I know that 0 and 2 are roots but, how do you solve it algebraically? or what theorem or method do you use to solve it? (without using numerical or graphical methods)

Thanks

I have to solve it with paper and pencil, I can't use calculator and I have to find exactly the roots 0 and 2.

Answer 1

1+4X=3^X

X=(3^X-1)/4

X can be only 0 and 2

Answer 2

I would use Newton Raphson method
1+4x=3^X

this is a nonlinear problem, so you can't solve it algebrically

let f(x) = 1 +4x - 3^x
so f'(x) = d/dx(1 +4x - 3^x) = 4 - 3^x ln(3)

assume any value, say x0 = 3, then compute

x1 = x0 - f(x0) / f'(x0)

then find,
x2 = x1 - f(x1) / f'(x1)
.....
xn1 = xn - f(xn) / f'(xn)

after say 7 or 8 steps you'll get very close to the answer.

Answer 3

1 + 4x = 3^x


try takin log.. n solv.

Answer 4

1+[4x3]
1+12
13

Answer 5

use natural logarithm theorem to solve it

Answer 6

you can use the Natural Log.