Question

10^x-8=√10^x. Type numerical part of answer.

Answer 1

By algebra substitution and algebra properties, the solution of the polynomial-like equation is: x₁ = 1.056, x₂ = 0.750.

How to solve a polynomial-like equation involving exponential expressions

In this problem we find the case of a polynomial-like equation, whose solution requires the employ of algebra substitution and algebra properties. First, define the entire equation:

`10^x - 8 = √(10^x)`

Second, use the following solution u² = 10ˣ:

u² - 8 = u

Third, solve the resulting polynomial:

u² - u - 8 = 0

Fourth, find the values of u by quadratic formula:

`u = (1 \pm √(1 - 4\cdot 1\cdot (- 8)))/(2\cdot 1)`

`u = (1 \pm √(33))/(2)`

u₁ ≈ 3.372 or u₂ ≈ - 2.372

Fifth, clear x by definition of logarithm:

Case 1: u₁ ≈ 3.372

10ˣ = 3.372²

10ˣ = 11.370

x = ㏒ 11.370

x₁ = 1.056

Case 2: u₂ ≈ - 2.372

10ˣ = (- 2.372)²

10ˣ = 5.626

x = ㏒ 5.626

x₂ = 0.750