find the solution set of logx+log(x+21)=2

Question

Logarithnic equation.

Answer 1

The logarithimic equation is log x + log ( x + 21 ) = 2.

Combine using the product rule of logarithms.

log x ( x + 21 ) = 2

Let both sides have a base of 10.

10^{log x ( x + 21 )}   = 10²

^{Remember that log is assumed to be 10 if there is no number specified.Because we made the equation have a base of 10 , we can cancel out the logs.} 

^{x ( x + 21 ) = 102} 

^{x ( x + 21 ) = 100}

x ² + 21 x  = 100

x ² + 21 x - 100 = 0

x ² + 25 x - 4 x  - 100 = 0

x ( x + 25 ) - 4 ( x + 25 ) = 0

( x + 25 )  ( x - 4 ) = 0

Apply product property.

x + 25 =0 ,   x -4  = 0

x  = - 25 ,  x = 4.

x = -25 is not valid

The solution set is 4.