What is the solution for log(3) x^2 + log(9) x = 2?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The equation to be solved is log(3) x^2 + log(9) x = 2

Use the following properties of logarithms

log a^b = b*log a , log a + log b = log a*b and log (b) c = log(a)c / log(a)b

log(3) x^2 + log(9) x = 2

=> log(3) x^2 + log(3) x / log(3) 9 = 2

=> log(3) x^2 + log(3) x / log(3) 3^2 = 2

=> log(3) x^2 + log(3) x^(1/2) = 2

=> log(3) x^2*x^(1/2) = 2

=> log(3) x^(5/2) = 2

=> x^(5/2) = 9

=> x = 9^(2/5)

=> x = 2.4082 ( approximately)

The solution of the equation is x = 2.4082

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