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

Question

Solve each equation. Give solutions in exact form. See Examples 5–9. log_5 [(3x + 5)(x + 1)] = 1

Accepted Answer

Hey, everyone. Here we are asked to solve the logarithmic equation for X and provide an exact solution here. The given equation is log base 14 of seven plus X plus log base 14 of two plus X is equal to log base 14 of 14. Here we have four answer choice options. Answer A, the solution set of negative nine and zero answer B, the solution set of zero answer C the solution set of negative nine and answer choice D no solution. So here to solve for X from our given equation, we want to begin by rewriting the equation by applying the product rule of logarithmic equations which tells us that log base A of X times Y is equal to log base A of X plus log base A of Y. So now applying this product rule to the left side of our equation, we can rewrite our equation as log base 14 of the quantity of seven plus X times the quantity of two plus X is equal to log base 14 of 14. So now the next step is to get rid of the log base 14 on both sides of the equation. And we can do this by having 14, be raised to the power of the left side of the equation and having 14 also be raised to the power of the right side of the equation. So now with this, we see that the 14 and the phase 14 on both sides of the equation cancel out. So now rewriting our expression, we have the quantity of seven plus X times the quantity of two plus X is equal to 14. Now continuing to solve for X, we can first mo supply the parentheses on the left side of our equation by applying the foil method. So we first multiply the first terms. So we have seven times two which gives us 14. Then multiplying the outside terms seven times X gives us plus seven X. Now taking the inside terms X times two gives us plus two X. And now multiplying the last terms X times X, we get plus X squared and this whole expression is still equal to 14. Now just combining like terms we have of X squared plus nine X plus 14 is equal to 14. Now moving the whole number from the right side to the left side by subtracting 14 from both sides of the equation. We are left with X squared plus nine X to zero. Now continuing to solve for X, we just want to factor out X from both terms on the left side of the equation. So we have X times the quantity of X plus nine is equal to zero. Now we can set both parts of the left side of the equation equal to zero individually. So we have X is equal to zero and we also have the expression X plus nine is equal to zero. So now we see that one solution that we have found directly is X is equal to zero. And now just solving the second expression here for X, we can subtract nine from. And we see that another possible solution is when X is equal to negative nine. But now with this negative nine solution, we want to plug this value back into the left side of our equation. And we see that we will have log base 14 of seven plus negative nine. And We find, we see that we get log base 14 of negative two and repeating for the other log logarithm expression. Here we have the log base 14 of 2 -9. And simplifying this gives us log base 14 of negative seven. And now we see that from substituting back in this negative nine value that we result in taking log base 14 of negative values which are undefined. So we know that X is equal to negative nine is not a solution. So here our only solution is when X is equal to zero. So this tells us that our solution set just contains zero. And so with this answer here we are left with answer choice B where we have the solution set of zero. 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. log_5 [(3x + 5)(x + 1)] = 1

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