solve equation log_{2}x + log_{x}2=2.5
First change the base on log_{x}2 using the change of base identity:
log_{x}2=log_{2}2/log_{2}x=1/log_{2}x
Then your equation becomes
log_{2}x + 1/log_{2}x =2.5
Let u=log_{2}x and simplify:
u+1/u=2.5
u²-2.5u+1=0
Use the quadratic formula to solve this quadratic equation, get u = 2 or 1/2.
For u=2=log_{2}x, x=4.
For u=1/2=log_{2}x, x=√2
Therefore, the two solutions are x=4 and x=√2
Check: log_{2}4 + log_{4}2 = 2+1/2=2.5
log_{2}√2 + log _{√2}2 = 1/2 + 2 =2.5