Solve each logarithmic equation in Exercises 49–92. Be sure to...

Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln(x−2)−ln(x+3)=ln(x−1)−ln(x+7)

Accepted Answer

Welcome back. I am so glad you're here. We are given this log rhythmic equation. The natural log of x minus five minus the natural log of X plus four equals the natural log of x minus two minus the natural log of X plus nine. And we are asked to solve for X. Look there are four, X is here and they're all inside these logs. What do we do? Well, we can recall from previous lessons on the expanding and condensing of logs that when we have two logs that are being subtracted from one another, that's the same as taking the log of these two parts that are inside the parentheses here and we're going to divide them. So the first one goes in the numerator X minus five and the second one that is being subtracted from the first one that goes in the denominator. So X plus four And we'll do the same thing on the right hand side, that will be the natural log of X -2 divided by X-us nine. And so now we can recall the 1 to 1 property of logarithms. Because we have a log rhythm, Rhythm on the left equal to a log rhythm on the right. We know that what's inside the parentheses are equal to each other. And we can just write that much. We can write x minus five divided by x plus four equals x minus two divided by x plus nine. And now look we've gotten them out of their logs and we can solve for X here. So let's do that. We will start by cross multiplying. So we've got X minus five times X plus nine equal to x minus two times X plus four. You might have those in a different order. That's okay. But you do need the X minus five times the X plus nine on one side and the X minus two times the X plus four on the other side. Now let's foil these out to get rid of our parentheses. First X times X on the left is X squared outside X times nine is plus nine X insides negative five times X is minus five. X lasts negative five times nine is minus equals on the right hand side, foiling firsts X times X is X squared outside X times four is plus four X insides negative two times X minus two X. And lasts negative two times four is minus eight. Great. Let's combine like terms in the middle on both sides of the equation. So on the left bring down the X squared nine X minus five X is plus four. X minus 45 equals Bring down X squared on the right, four X minus two, X is plus two X. Bring down that minus eight. Okay, now let's bring all of the X is to the left hand side of the equation. Will subtract X squared from both sides and will subtract two X from both sides and let's bring this 45 negative 45 to the right hand side of the equation will add 45 to both sides. So on the left the X squared cancel out four X minus two X. Is two X. And the 40 five's cancel out equals that equals but it looks kind of like their equals. And on the right hand side X squared cancel out two X's cancel out. And we have negative eight plus 45 which is 37. We can divide both sides by two and we have X equals 37 halves. Or as a decimal 18.5 or 50500. But 18.5 works for this next part, which is to find out whether we are within the constraints of our domain. So when we have logs anything inside of our log in these parentheses here needs to be greater than zero. It needs to be positive. So we need to check all of these. Let's check and see if 18.5 plugged in for X will give us positive values. So we have 18.5 minus five, 18.5 Plus four, 18.5 - And 18.5-plus 9. All four have to be positive values for 18.5 to work as a solution. Well 18.5 minus five is 13.5, Nice 18.5 plus four is 22.5 Nice 18.5 minus two is 16.5 yes and 18.5 plus nine is 27.5. Yeah. Alright, 18.5 works. So we look at our answer choices and this matches with answer choice C. Well done, we'll catch you on the next one.

Key Concepts

Properties of Logarithms

Domain of Logarithmic Functions

Solving Logarithmic Equations

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