Question

Log(3)(x-5)+log(3)(x+2)=3

Answer 1

Log3(x-5)+log3(x+2)=3
ОДЗ:
{x-5>0   {x>5   ⇒ x>5
{x+2>0   {x>-2

log3((x-5)(x+2))=log3(3^3)

x^2+2x-5x-10=27
x^2-3x-37=0
D=(-3)^2-4*1*(-37)=157
x1=(3-√157)/2  - не удовлетворяет ОДЗ
x2=(3+√157)/2 

Answer 2

To solve this logarithmic equation, we can use the following property of logarithms:

log base a (mn) = log base a (m) + log base a (n)

Using this property, we can combine the two logarithms on the left-hand side of the equation:

log(3)((x-5)(x+2)) = 3

Now, we can rewrite this equation in exponential form:

3^3 = (x-5)(x+2)

27 = x^2 - 3x - 10

0 = x^2 - 3x - 37

We can now solve for x using the quadratic formula:

x = (3 ± sqrt(3^2 - 4(1)(-37))) / (2(1)) x = (3 ± sqrt(145)) / 2

Therefore, the solutions to the equation are:

x = (3 + sqrt(145)) / 2, or x = (3 - sqrt(145)) / 2

However, we need to check these solutions to make sure they are valid. We can do this by substituting each solution back into the original equation and verifying that both sides are equal.

Substituting x = (3 + sqrt(145)) / 2:

log(3)((3 + sqrt(145))/2 - 5) + log(3)((3 + sqrt(145))/2 + 2) = 3

This equation is true, so x = (3 + sqrt(145)) / 2 is a valid solution.

Substituting x = (3 - sqrt(145)) / 2:

log(3)((3 - sqrt(145))/2 - 5) + log(3)((3 - sqrt(145))/2 + 2) = 3

This equation is not true, so x = (3 - sqrt(145)) / 2 is not a valid solution.

Therefore, the only solution to the equation is:

x = (3 + sqrt(145)) / 2.