Solve each equation. Give solutions in exact form. See Examples 5...

Question

Solve each equation. Give solutions in exact form. See Examples 5–9. ln x + ln x^2 = 3

Accepted Answer

Hey, everyone here we are asked to solve the logarithmic equation for X and provide an exact solution here. Our given equation is the natural log of X raised to the 11th power plus the natural log of X is equal to 12. Here we have four answer choice options, answer choice. A, the solution set of E answer B, the solution set of one answer C, the solution set of one and E and answer choice D, the solution set of three and E Looking at our given logarithmic equation, we see that our first term is the natural log of X raised to the 11th power. Well, with this term, we can recall that the natural log of X raised to the power of a is equivalent to a times the natural log of X. So with this statement here, we can rewrite the first term in our given equation as 11 times the natural log of X plus the natural log of X is equal to 12. So now we can just combine our like terms. We see we have 11 times the natural log of X plus the natural log of X which gives us 12 times the natural log of X is equal to 12. Now getting rid of the coefficient of the natural log, we can divide both sides by 12. And we see we end up with the natural log of X is equal to one. Now to continue to manipulate this equation, we need to recall that the natural log of X is equivalent to log base E of X. So now we can rewrite our equation where we have the natural log of X is equal to one as log base E of X is equal to one. Now, we can isolate our variable X by recalling that Y is equal to log base A of X can be rewritten or is equivalent to X is equal to A to the power of Y. So now instead of having log base E of X is equal to one, we can rewrite this in the second form or the equation where X is equal to A, to the power of Y. So we will have X is equal to E E to the power of one. Well, E to the power of one is just E. So we actually have X is equal to E and therefore we see that we have our solution of E or we have the solution set of just E. So with our answer here, we are left with answer choice. A where again we have the solution set of E. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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Solve each equation. Give solutions in exact form. See Examples 5–9. ln x + ln x^2 = 3

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