How do you solve for x in log(5−x)−13log(35−x3)=0?

1 Answer
George C.
Dec 6, 2015

Rearrange and derive a quadratic equation, one of whose roots is a valid solution of the original problem:

x=75−3√10526≈1.702

Explanation:

Add 13log(35−x3) to both sides to get:

log(5−x)=13log(35−x3)

Multiply both sides by 3 to get:

log(35−x3)=3log(5−x)=log((5−x)3)

Since log is one-one (as a Real valued function), we must have:

35−x3=(5−x)3=53−3(52)x+3(5)x2−x3

=125−75x+13x2−x3

Add x3 to both sides to get:

13x2−75x+125=35

Subtract 35 from both sides to get:

13x2−75x+90=0

Use the quadratic formula to find:

x=75±√752−4⋅13⋅902⋅13

=75±√94526

=75±3√10526

We need to check these solutions for validity:

If x=75+3√10526≈4.067 then x3>64, so 35−x3<0 and log(x) is not well defined.

If x=75−3√10526≈1.702 then 5−x>0 and 35−x3≈35−4.93>0, so both logs are well defined Real valued.

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