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Solve each inequality for x.
$$(a)\ln x<0 \quad \text { (b) } e^{x}>5$$
(a) $0< x<1$(b) $x>\ln 5$
Calculus 1 / AB
Chapter 1
Functions and Sequences
Section 5
Logarithms; Semilog and Log-Log Plots
Functions
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lawn of X is less than zero. That is the inequality you want to try to solve. So when it says this solving a call whenever you see that instruction, regardless of what the inequality is you're trying to isolate X, you're trying to get it by itself. And so in order to do that, we have to remove this lawn. So what we have to think about is what is the inverse operation of lawn? How do how do I remove the lawn? And the way you do that is by using this rule of law algorithms, which says that if you have log of some base B of a number, let's say a and it's equal to some number. See that you can rewrite that statement. So if you have a log on one side and a number on the other, you can rewrite as an exponential, and what you do is you take the base, so whatever that base happens to be, we're gonna call it base is be for now. But it's always gonna be some number, and in our case, the natural log rhythm is based E, and then you raise it to the exponents So you say B to the sea. You kind of think about it this way is equal to a So in order to remove the lager them in this question. What we're gonna do is we're just gonna rewrite it as e, which is our base raised to that exponents zero, and that's going to give us the number. And then we simply figure out what is eating zero on evens. Here was one. So our answer from removing the logger them is axe is less than one. But that's not quite the final answer because we have to consider lawn or any log rhythm is not allowed to be negative. And the reason is not a levee negative. You think graphically, longer them's looked like this. They have a vertical ash into their they don't ever cross zero. So you kind of think photographically you can see why negatives might be a problem. And if you think about it in this context, in this rule that we just talked about, you can also see why negatives to be a problem, I think, because if you take the base and it's positive and all the bases are that we use longer than is for. And no matter what you raise it to, then it's an exponents. You can never get a negative number from a positive base, Raceman exponents. You can't do that mathematically, and so we can't have negative. So we have to add in a stipulation, saying X has to also be greater than zero. And that's my solution. The lawn X is less than zero. It's remove the wog rhythm by writing as an exponential men solving. Now, the second part of this question, part beast, is e to the X is greater than five, and I want to know the solution to that. So it's a slightly different take on the same rule we know we just wrote down that log base B of A is equal to see means that hey is beat of the sea. So in the first part, in part A. We used this to this. We went from the logger them to the exponential. In this question, we're actually gonna go from the exponential to the logger them. So the base is E. This is my exponents X, and that's the number that I end up with. That's the value so what I'm gonna do. He is. I'm gonna rewrite this now as a log rhythm. So this is the exponents of X becomes greater than log of the base, which is e and then of that number five. So we just then can write. This log base is written as a natural longer than it has its own statement as lawn five. So my answer is X has to be greater than lawn five. So we're using the same rule in the first case just to quickly recap We took the log rhythm and wrote it as an exponential to solve this. And in the second case, we took the exponential and Rodas of lager than to solve it.
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