Solve for x:

Question

log₄x(2x+7) +log₄ (x) = 1? 

Answer 1

The logarithmic equation is log₄(x)(2x + 7) + log₄(x) = 1.

Apply product property of logarithm : .

log₄(x)(x)(2x + 7) = 1

log₄(x²)(2x + 7) = 1

log₄(2x³ + 7x²) = 1

According to basic logarithmic rule.

2x³ + 7x² = 4¹

2x³ + 7x² = 4

2x³ + 7x² - 4 = 0.

Graph :

The points where it crosses the x  axis  will give solutions to the polynimial function .

The graph crosses the x  - axis at a point that would suggest a factor.

It crosses the x  - axis at one point hence there are one real root.

x  = 0.677.

Use synthatic division to detrmine if the given value of x  is a root of the polynomial.

0.677 | 2     7    0     - 4

         | 0  1.35   6.42    4

         ______________________

              2    9.48  6.42   0

Since f (0.677) = 0, x = 0.677 is a zero.

The depressed polynomial is  2x² + 9.48x + 6.42 = 0.

Since the depressed polynomial of this zero, 2x² + 9.48x + 6.42 = 0, is quadratic,

use the Quadratic Formula to find the roots of the related quadratic equation

 

a = 2, b = 9.48, and c = 6.42.

x = [- 9.48 ± √(9.48² - 4 * 2 * 6.42)]/2 * 2

x = [- 9.48 ± √(89.9 - 51.4)]/4

x = [- 9.48 ± √38.5]/4

x = [- 9.48 ± 6.204]/4

x = - 0.8187 and x = - 3.42

Therefore, the solutions of the given quadratic equation are x = - 0.8187, x = - 3.42, and x = 0.677.