How to solve the equation log_(x) 5+log_(5) x=5/2 ?

1 Answer
Jan 25, 2018

x_1=sqrt5 and x_2=25

Explanation:

log_x5+log_5x=5/2

log_x5+1/log_x5=5/2

After using y=log_x5, this equation became

y+1/y=5/2

(y^2+1)/y=5/2

2*(y^2+1)=5*y

2y^2+2=5y

2y^2-5y+2=0

(2y-1)*(y-2)=0

From this equation, y_1=1/2 and y_2=2

For y=2, log_x5=2 or x^2=5. So x_1=sqrt5

For y=1/2, log_x5=1/2 or x^(1/2)=5. So x_2=5^2=25