SOLUTION: Solve the equation {{{2^(x^3)=3^(x^2)}}}

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Question 143994: Solve the equation 2%5E%28x%5E3%29=3%5E%28x%5E2%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
2%5E%28x%5E3%29=3%5E%28x%5E2%29 Start with the given equation


log%2810%2C%282%5E%28x%5E3%29%29%29=log%2810%2C%283%5E%28x%5E2%29%29%29 Take the log of both sides


x%5E3%2Alog%2810%2C%282%29%29=x%5E2%2Alog%2810%2C%283%29%29 Rewrite the expressions using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


x%5E3%2Alog%2810%2C%282%29%29-x%5E2%2Alog%2810%2C%283%29%29=0 Subtract x%5E2%2Alog%2810%2C%283%29%29 from both sides


x%5E2%28x%2Alog%2810%2C%282%29%29-log%2810%2C%283%29%29%29=0 Factor out the GCF x%5E2


x%5E2=0 or x%2Alog%2810%2C%282%29%29-log%2810%2C%283%29%29=0 Set each individual factor equal to zero


Let's solve the first equation x%5E2=0


x=0 Take the square root of both sides. So this is our first answer.

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Now let's solve the second equation x%2Alog%2810%2C%282%29%29-log%2810%2C%283%29%29=0

x%2Alog%2810%2C%282%29%29=log%2810%2C%283%29%29 Add log%2810%2C%283%29%29 to both sides


x=log%2810%2C%283%29%29%2Flog%2810%2C%282%29%29 Divide both sides by log%2810%2C%282%29%29 to isolate x


x=log%282%2C%283%29%29 Use the change of base formula to simplify. So this is our other answer.



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Answer:

So the solutions are
x=0 or x=log%282%2C%283%29%29

which approximate to

x=0 or x=1.58496