solvefor x,e^{x/2}=4lnx

解答

求解

x, e^{\frac{x}{2}} = 4 ln (x)

x \approx 1.92309, 2.89429

求解步骤

e^{\frac{x}{2}} = 4 ln (x)

Find the difference function

f (x) = e^{\frac{x}{2}} - 4 ln (x)

Look for the function's extreme points

e^{\frac{x}{2}} - 4 ln (x), x > 0

的极值点:

极小值

(2.40433 \dots, - 0.18177 \dots)

Check function behavior at domain edges:

0, \infty

x \to 0 + lim (e^{\frac{x}{2}} - 4 ln (x)) = \infty

x \to \infty lim (e^{\frac{x}{2}} - 4 ln (x)) = \infty

The interval between every consecutive point pair is monotone

x f (x) 0 \infty 2.40433 \dots - 0.18177 \dots \infty \infty

Check every interval, if the signs at the edges are different, find the X intersection using Brent method

Between

(0, \infty)

and

(2.40433 \dots, - 0.18177 \dots)

1.92309

Between

(2.40433 \dots, - 0.18177 \dots)

and

(\infty, \infty)

2.89429

x \approx 1.92309, 2.89429

2^{- b} = 1

5^{5 - x} = \frac{1}{25}

so l v e f or x, 312 5^{x + 4} = 2 5^{2 x}

7^{y} = 15

3^{2 x} - 3 (3^{x}) = 4