Question

42. Find all solutions to the following equations. Remember to check for extraneous solutions.

Answer 1

Answer:

The solution of the equation is 5.85

Step-by-step explanation:

Given log(x − 1) + log(x− 4) = 1, you have to apply logarithm properties to find the solution:

1) log(x − 1) + log(x− 4) = 1 As the sum of logarithm of the same base, is equal to solve the logarithm of the product, then

                      log(x − 1) + log(x− 4) = 1 ⇒ log [(x- 1)(x -4)] = 1

                       ⇒ log (x² -4x - x + 5) = 1

2) The following property tha you have to apply states tha

log x = b ⇒ x = 10^b

⇒ log (x² -4x - x + 5) = 1 ⇒ x² -4x - x + 5 = 10^1 ⇒ x² -4x - x + 5 = 10

Reordering..... x² -4x - x + 5 -10 = 0 ⇒ x² -4x - x - 5 = 0

At this point you can solve the equation by finding tghe roots of a quadratic equation.

⇒ x = ( -b ± √[b² -4ac])/ 2a where a is the quadratic coefficient (1), b the linear coefficient (-5) and c the independent coefficient (-5).

⇒ x = ( 5 ± √[5² -4(1)(-5)])/ 2(1) ⇒ x= ( 5 ± √45)/2

You have two solutions x = 5.85 and x = -0.85

To check this solutions you have to replace it in the original equation. As

log(5.85 − 1) + log(5.85− 4) = 1 ⇒ 1=1. It is correct.

log(-0.85 − 1) + log(-0.85− 4) = 1, Here as you know, the logarithm of a negative value is not defined so, this is not a solution.

Summarizing, the solution of the given equation is 5.85.