Solve 2^x=x^2+7 | Find x Without Differentiation

[Harmony]

2^x=x^2+7
Find x.

The answer is 5. I have tried substituting 2^x=a, but that doesn't help. Is differentiation method required here? If not, how should I approach the question?

 

[dextercioby]

The equation is awfully transcendetal. My advice is to find the solution by a graphical method.

$$x=\frac{1}{\ln 2}\ln\left(x^{2}+7\right)$$

Daniel.

 

[Steve10]

2^x = x^2 + 7
2^x - x^2 - 7 = 0

Now use the http://www.shodor.org/UNChem/math/Newton/index.html" to find root(s) of f(x) = 0

If the eqn, 2^x=x^2+7, looks simple enough you could try simple substitution.

x=1: 2 = 1 + 7 -- nope, lhs too small
x=2: 4 = 4 + 7 -- nope, lhs too small
x=3: 8 = 9 + 7 -- nope, lhs too small
x=4: 16 = 16 + 7 -- nope, lhs too small
x=5: 32 = 25 + 7 -- yep, lhs = rhs

If the above substitution method didn't work, and you went from lhs too small to rhs too large, with a value of x = xo, say, then you could try the http://www.shodor.org/UNChem/math/Newton/index.html" with xo as your first approximation.