SOLUTION: solve exactly (lnx)^3= lnx^9

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Question 110930: solve exactly
(lnx)^3= lnx^9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let w=ln%28x%29

So we get w=ln%28x%5E9%29

w%5E3=9%2Aln%28x%29 Rewrite the logarithm using the identity ln%28x%5Ea%29=a%2Aln%28x%29


w%5E3=9%2Aw Replace ln%28x%29 with w


w%5E3-9%2Aw=0 Subtract 9w from both sides

w%28w%5E2-9%29=0 Factor out a w



w%28w%2B3%29%28w-3%29=0 Factor using the difference of squares


So our solutions are (in terms of w)

w=0, w=-3, or w=3


So this means ln%28x%29=0, ln%28x%29=-3, or ln%28x%29=3


Now raise all sides as an exponent with base e


e%5E%28ln%28x%29%29=e%5E0, e%5E%28ln%28x%29%29=e%5E%28-3%29, or e%5E%28ln%28x%29%29=e%5E3

Simplify

x=1, x=1%2Fe%5E3, or x=e%5E3