Solving logarithmic equation, different bases What number do I need to multiply both sides with? I have worked for an hour on this but it is the first time I am using this website so it is impossible for me to write what I have already done. If you can give me tips on how to solve it I would appreciate it a lot! 2 log_6 (sqrt(x) + root[4](x)) = log_4 x
Uroskopieulm
Answered question
2022-11-17
Solving logarithmic equation, different bases What number do I need to multiply both sides with? I have worked for an hour on this but it is the first time I am using this website so it is impossible for me to write what I have already done. If you can give me tips on how to solve it I would appreciate it a lot!
Answer & Explanation
sellk9o
Beginner2022-11-18Added 11 answers
Note that for any and , so that we can rewrite this as
From here, the easiest thing is to guess, and observe that for the argument of the left-side log comes out nicely if is a fourth power. For instance, we might guess , which yields , and indeed
ETA: That this is the only solution for can be seen if we write (following GEdgar's comments to the original post)
or equivalently
Note that the argument of the is a decreasing function of , which means that the entire right-hand side is a decreasing function of for , which in turn means that the solution is the only solution for For , the argument of the is greater than , but its base is less than , so the log is negative, and the right-hand side is less than , whereas the left-hand side is greater than , so there are no solutions in that range, either. Finally, is not a solution of the original equation, by inspection.