How do you solve #3 log x = 6 - 2x#?

1 Answer
Costas C.
Jan 13, 2016

Not sure if it can be solved
If you are really curious about the number, the answer is:

#x=2.42337#

Explanation:

Other than using Newton's method, I am not sure if it is possible to solve this. One thing you can do is prove that it has at exactly one solution.

#3logx=6-2x#
#3logx+2x-6=0#

Set:

#f(x)=3logx+2x-6#

Defined for #x>1#

#f'(x)=3/(xln10)+2#

#f'(x)=(3+2xln10)/(xln10)#

For every #x>1# both the numerator and denominator are positive, so the function is increasing. This means it can only have a maximum of one solution (1)

Now to find all the values of #f(x)# #x>1# means #x in(0,oo)#:

#lim_(x->0^+)f(x)=lim_x->(0^+)(3logx+2x-6)=-oo#

#lim_(x->oo)f(x)=lim_(x->oo)(3logx+2x-6)=oo#

Therefore, #f(x)# can take any real value, including 0, which means that #f(x)=0<=>3logx+2x-6=0# can be a solution at least once (2)

(1) + (2) = (Maximum of one) + (At least one) = Exactly one

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