Question

Log₂ (x+5)-1og₂ (x-2)=3

Answer 1

Answer: x=3

Explanation:

log₂(x+5)-log₂(x-2)=3

Domain:

x+5>0 ⇔ x>-5

x-2>0 ⇔ x>2

Hence, x>2 or x∈(2,∞)

Solution:

`\displaystyle\\log_2(x+5)/(x-2) =3log_22\\\\log_2(x+5)/(x-2) =log_22^3\\\\log_2(x+5)/(x-2) =log_28\\\\Hence,\\\\(x+5)/(x-2)=8`

Multiply both parts of the equation by (x-2):

`x+5=8(x-2)\\\\x+5=8x-16\\\\x+5-x=8x-16-x\\\\5=7x-16\\\\5+16=7x-16+16\\\\21=7x`

Divide both parts of the equation by 7:

`3=x\\\\Thus,\ x=3`