Question

Hello could you help me solve this question? (9.11 Solving Logarithmic Equations #14)

Answer 1

`x_(1)=(-3+33)/(2)`

Step-by-step explanation

Given

`2\log_{(1)/(2)}(x+2)=\log_{(1)/(2)}(x+10)`

We can use the logarithmic properties to simplify our equation, where:

`a\log_bx=\log_bx^a`

`a^(\log_a(x))=x`

Applying this rule to both sides of our problem given:

`(x+2)^2=(x+10)`

Solving the squared expression:

`x^2+4x+4=x+10`
`x^2+4x-x=10-4`
`x^2+3x=6`

Setting the equation to 0:

`x^2+3x-6=0`

Using a scientific calculator to get the result:

`x_1=(-3+√(33))/(2)`
`x_1=(-3-√(33))/(2)`

If we verify the solutions the correct one is the first.