Question

Solve the following logarithmic equations
log(x + 3) + log(x + 5) = 2

Answer 1

x² + 8x - 85 = 0

and, the roots are x =
`-4+√(101),\:x=-4-√(101)`

Explanation:

Data provided:

log(x + 3) + log(x + 5) = 2 ..........(1)

Now,

From the properties of the log function, we know

log(A) + log(B) = log(AB)

therefore,

applying the above property in the equation (1), we get

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

or

log(x² + 3x + 5x + 15) = 2

taking the anti-log both sides, we get

x² + 3x + 5x + 15 = 10²

or

x² + 8x + 15 = 100

or

x² + 8x + 15 - 100 = 0

or

x² + 8x - 85 = 0

on solving the above equation

the roots are

x =
`-4+√(101),\:x=-4-√(101)`