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_3 [(x + 5)(x - 3)] = 2

Accepted Answer

Hey, everyone. Here we are asked to solve the logarithmic equation for X and provide an exact solution. Here, we are given the equation log of 52 plus X is equal to log of six plus log of X plus seven. Here we have four answer choice options. Answer A, the solution set of two, answer B, the solution set of negative two answer C the solution set of negative two and four and answer D no solution. So looking at our given logarithmic equation in order to solve this equation for X, we first need to apply the product property of logarithm to the right side of our equation. So if we recall the product property of logarithms, we know that log base A of X times Y is equivalent to log base A of X plus log base A of Y. So now again, applying this rule to the right side of our given equation, we can rewrite our equation as log of 52 plus X is equal to log of six times the quantity of X plus seven. So now with our rewritten equation, we see that both sides of the equation have a logarithm of some expression. And so we want to get rid of the logarithm. And to do this, we need to have 10, be raised to the power of the left side expression. And we also need 10 to be raised to the power of the expression on the right side. So doing this, we see that the 10 and the logarithms both cancel. So now rewriting, rewriting our equation, we have 52 plus X is equal to six times the quantity of X plus seven. So now we just need to manipulate this equation to solve for X. So we first just want to factor in this six on the right side of the equation. So we have 52 plus X is equal to six X plus 42. And so now we want to get our, our whole numbers on the same side of the equation. So we can subtract 42 from both sides. And we get that 10 plus X is equal to six X. Now getting the X terms on the same side, we can subtract X from both sides and we get 10 is equal to five X. Now getting rid of the coefficient of X, we divide both sides by five. And we see that the solution or the value for X is two. And so since X is equal to two, we see that the solution set for our given equation here just has positive two. And therefore we are left here with answer choice. A where the solution set is of two. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation. Give solutions in exact form. See Examples 5–9. log_3 [(x + 5)(x - 3)] = 2

Key Concepts

Logarithmic Functions

Properties of Logarithms

Solving Quadratic Equations

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