In Exercises 93–102, solve each equation. 3|log x|−6=0

Question

Accepted Answer

Welcome back. I am so glad you're here. We are given this equation, we're told that seven times the absolute value of the log of x minus seven equals zero. And we are to solve for X in here. Well there are a number of steps to get X out of all of those things. But let's start off by adding seven to both sides, scooting this seven over. So now we've got seven times the absolute value of the log of X equals seven. Now we'll divide both sides by seven and the seven's cancel on the left. So we've got the absolute value of the log of X equals seven divided by seven is one. Okay, now this is great because the absolute value equals a positive value, positive or equal to zero. So that's great. We can continue and we can recall from previous lessons on absolute value. That absolute value is the distance that something is from zero on the number line. So the log of X is one unit away from zero, but it's one unit away from zero in both the positive and the negative direction. So we can set two equations, we can say the log of X is equal to negative one and the log of X is equal to positive one. And now we're really close to getting X by itself. But how do we get it out of that log? Will we recall from previous lessons that we know two different equations for log in the way that it relates to exponents. We know that why equals log B X. And that's really similar to what we have here, we have the log of X equals a number. And we also know that be raised to the Y equals X. And that's great because this doesn't have a log in it and it has an X. By itself. So how do we get from one equation to the other? Well, let's identify all of our parts are Y and R. Two equations. Well, that's what's on the opposite side of the equal sign from the log B of X. So in this case the Y is going to be a negative one And a positive one. We've identified our wise so that one's done, log is just log. What are our bees? Alright. There are no little numbers down here below the log. And when that's the case, then it's understood that the log is 10. I'll rewrite that up here. B equals 10. All right, we found our B and X. That's what we're trying to solve for. So rewriting as this second equation, we're going to write be raised to the Y. And that'll equal R. X. So on the left hand side, B is 10 raised to the Y. Is negative one that equals x. 10 raised to the negative one. That is the same as 1/10. So x equals 1 10th over here. Be raised to the Y. So B is 10 raised to the Y. Y. Is one equals X. Okay. 10 to the first. That's just 10. So we have X equals 10. Okay. We've solved for R two X values and with logs, we just need to make sure that they're both greater than zero for our domain and yes, they are both greater than zero. So we look at our answer choices and this matches with answer choice. C. Well done, we'll catch you on the next one.

In Exercises 93–102, solve each equation. 3|log x|−6=0

Key Concepts

Logarithmic Functions

Absolute Value

Solving Equations Involving Logarithms and Absolute Values

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