In Exercises 93–102, solve each equation. 2|ln x|−6=0

Question

Accepted Answer

Welcome back, I am so glad you're here. We are given the equation five times the absolute value of the natural log of X -25 equals zero and we are asked to solve for X. Well X is kind of buried in there between a lot of things. But let's start off by moving 25 over to the right hand side of the equation will add 25 to both sides. So we'll have five times the absolute value of the natural log of X equals positive. 25 will divide both sides by five to get that absolute value by itself and so on the left will have the absolute value of the natural log of X. And on the right hand side, 25 divided by five is five. Okay, the absolute value equals a positive amount. So that's great. No domain restriction there. We can proceed and we recall from previous lessons on absolute value that the absolute value is the distance that something is from zero on the number line. So in this case the natural log of X is five units away from zero and it's not only five positive units away from zero, it's five units from zero in the negative direction as well. So we can write two equations. We can say that the natural log of X is equal to negative five and the natural log of X is equal to positive five. And now we're so close to getting X by itself. But how do we get it out of that? Natural log? Well, we recall from previous lessons that we know a couple of equations about natural log and how it relates to exponents. We know that Y equals the natural log base E. Of X. And that E raised to the Y equals X. And what we've got here looks a lot like this first equation where the natural log of X is equal to a number. And we want to get it to look like the second equation because there's no natural logs there, which is lovely. And then the X is by itself on one side. So that's what we want. How do we get from one to the other? Well, let's identify all of our different parts. What is our why? Well, why is the opposite side of the equal sign as the natural log of X. So in this case why equals negative five And positive five. OK. We figured out our why natural log that just is what it is, that E that's just understood to be there. E is E. It's a constant and X is what we're looking for. So now we can rewrite this as E raised to the Y. So over here, Y is negative five equals X. And on the right hand side this will be E raised to the Y, which is five equals X. Well, this one E to the fifth equals X. That one's as simplified as it's going to get, we can do a little bit more simplification with the race to the negative fifth. We'll get that negative out of the exponents, so we'll turn this into one over E. To the fifth equals X. And then we pause and ask ourselves if these are both positive because we have to have positive natural logs and logs and yes, these are both positive values are greater than zero, so they work. And we look at our answer choices and these match with answer choice. C. Well done, we'll catch you on the next one.

In Exercises 93–102, solve each equation. 2|ln x|−6=0

Key Concepts

Logarithmic Functions

Absolute Value

Solving Exponential Equations

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