Question

What is the true solution to the logarithmic equation below?
log₂ (6x)-log₂ (√x)=2

Answer 1

answer: x =
`\\ (4)/(9\\)`

`log_(2)` (6x) -
`log_(2)` (
`√(x)`) = 2

expand the expression:

• 
  `log_(2)`(6x) =
  `log_(2)` (6) +
  `log_(2)` (x)

• 
  `log_(2)` (
  `√(x)`) =
  `log_(2)` (
  `x^{(1)/(2) }`)

transform the expression:

• 
  `log_(2)` (
  `x^{(1)/(2) }`) =
  `(1)/(2)` *
  `log_(2)` (x)

calculate the difference:

• 
  `log_(2)` (x) -
  `(1)/(2)` *
  `log_(2)` (x) =
  `(1)/(2)` *
  `log_(2)` (x)

multiply both sides by 2:

• 
  `log_(2)` (6) +
  `(1)/(2)` *
  `log_(2)` (x) = 2 is now 2
  `log_(2)` (6) +
  `log_(2)` (x) = 4

transform the expression:

• 2
  `log_(2)` (6) =
  `log_(2)` (
  `6^(2)`)

simplify the expression:

• 
  `log_(2)` (
  `6^(2)`) +
  `log_(2)` (x) =
  `log_(2)` (
  `6^(2) x`)

evaluate the power:

• 
  `log_(2)` (
  `6^(2) x`) =
  `log_(2) (36x)`

convert the logarithm into exponential form:

• 
  `log_(2)(36x) = 4` is now
  `36x = 2^(4)`

evaluate the power:

• 
  `2^(4) = 16`

divide both sides by 36:

• 
  `(36)/(36)` x =
  `(16)/(36)` which is x =
  `(4)/(9)`