If log (x + 2) + log (x − 2) = log 5 then the value of x will be

Concept:

Logarithm properties

Product rule: The log of a product equals the sum of two logs.

\({\log _{\rm{a}}}\left( {{\rm{mn}}} \right) = {\rm{\;}}{\log _{\rm{a}}}{\rm{m}} + {\rm{\;}}{\log _{\rm{a}}}{\rm{n}}\)

Quotient rule: The log of a quotient equals the difference of two logs.

\({\log _{\rm{a}}}\frac{{\rm{m}}}{{\rm{n}}} = {\rm{\;}}{\log _{\rm{a}}}{\rm{m}} - {\rm{\;}}{\log _{\rm{a}}}{\rm{n}}\)

Calculation:

Given:

log (x + 2) + log (x − 2) = log 5

⇒ log [(x + 2) (x – 2)] = log 5                (∵ \({\log _{\rm{a}}}\left( {{\rm{mn}}} \right) = {\rm{\;}}{\log _{\rm{a}}}{\rm{m}} + {\rm{\;}}{\log _{\rm{a}}}{\rm{n}}\))

⇒ log (x ² – 4) = log 5

⇒ x² – 4 = 5

⇒ x² = 9

∴ x = ± 3

Here x ≠ -3 is not possible because (x – 2) should be greater than zero.

∴ x = 3