Question

Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. log Subscript 5 Baseline left parenthesis x plus 22 right parenthesis minus log Subscript 5 Baseline left parenthesis x minus 2 right parenthesis equals 2

Answer 1

Value of x = 3

Explanation:

Given a logarithmic function:

⇒
`log_5\ (x+22) - log_5\ (x-2) =2`

⇒
`log_5\ (x+22) =2+log_5\ (x-2)`

...subtracting log_5(x-2) both sides.

⇒
`log_5\ (x+22) =log_5\ (5^2)+log_5\ (x-2)`

Using log of the base
`log_5(5) = 1` and
`x=log_y(y^x)` so
`2=log_5(5^2)`

⇒
`log_5\ (x+22) =log_5\ (25)+log_5\ (x-2)`

Applying log product rule ...
`log_x(a)+log_x(b)=log_x(ab)`

⇒
`log_5\ (x+22) =log_5\ 25(x-2)`

⇒
`(x+22) =25(x-2)`

⇒
`x+22=25x-50`

⇒
`25x-50-x-22=0`

⇒
`25x-x-50-22=0`

⇒
`24x=72`

⇒
`x=(72)/(24)`

⇒
`x=3`

The value of x in the logarithmic equation, = 3