Logarithm expression

Question

Log5x+3log5x=2

Answer 1

Apply the rules of logarithms.  When adding logs, we multiply the arguments of the logs.  Also, the coefficient of the log becomes of the exponent of the argument.

 

 

log₅(x) + log₅(x³) = 2

 

log₅(x * x³) = 2

 

log₅(x⁴) = 2

 

 

The solution to a log is the exponent of the log's base number.

 

x⁴ = 5²

 

x⁴ = 25

 

 

Subtract 25 on both sides of the equation.

 

x⁴ - 25 = 0

 

 

Noticing this is a difference in perfect squares.

 

(x² - 5)(x² + 5) = 0

 

 

x² - 5 = 0        ----->    real factor

 

x² = 5

 

x = ±√5

 

 

The only real solutions we can have are

 

x = -√5           or             x = √5

 

 

We accept    x=√5    as a solution, since the argument cannot be negative.