Question

Solve the equation for x, accurate to three decimal places: 

Answer 1

The exponential equation is e^x - √e^x = 6

e^x - √(e^x) - 6 = 0

Rewrite the equation in quadratic form.

[√e^x]²- √(e^x) - 6 = 0

Factor the left side of equation.

(√e^x)² - 3√(e^x) + 2√(e^x) - 6 = 0

√(e^x)[√(e^x) - 3] + 2[√(e^x) - 3] = 0

[√(e^x) - 3] [√(e^x) + 2] = 0

Apply product rule.

[√(e^x) + 2] = 0 and [√(e^x) - 3] = 0

√(e^x) = -2 and √(e^x) = 3

Squre root value is always positive.

When √(e^x) = -2, No solution exist.

So consider √(e^x) = 3

Take the natural log of both sides:

ln(√(e^x)) = ln(3)

ln(e^x)^1/2 = 1.098

From exponents rules (a^x)^y = a^xy

ln(e^x/2) = 1.098

Apply the power property of logarthim log_bA^C = C(log_bA)

(x/2) ln(e) = 1.098

Since ln(e) = 1

x/2 = 1.098

x = 2(1.098)

x = 2.196

Solution x = 2.196.