Explain why or why not. Determine whether the following statem...

Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

b. ln(x + 1) + ln(x − 1) = ln(x² − 1), for all x.

Accepted Answer

Welcome back, everyone. In this problem, for which values of X is the equation LNX2 minus 4 X + 4, plus LNX2 + 2 X + 1 equal to LNX minus 2 are multiplied by X + 1 R22. A says it's true for the values of X that are all real numbers except when it's -2 and 1. B says it's true except when X is 1 and 2. C says it's true except when X is -2 and 1 and D says it's true except when X is either 1 or 2. Now, for us to figure out which values makes this equation true, let's first check that the identity is true and then we can test which values are going to work, OK? Now how can we know if this is true? We'll recall the properties of logarithms for the sum rule of two logarithms with the same base. We know that if we're adding two logs, so let's say log AFF X. Plus log BFF X. Log A, sorry, Log A G of X, my apologies here. Like a GRX. Then It's going to be equal to the log base A of FG of X. So the question we have to ask ourselves is, can we show that our left-hand side is equal to our right hand side using this property? Well, 4 X2 for a first term X2 minus 4 X + 4. Not that it is a perfect square, which can be written as X minus 2 or 2, OK? And for our next term, X2 + 2 X + 1 or 2 polynomial, it is also a perfect square that can be written as X + 1 squared. So this tells us then that for our left hand side we could rewrite it as LNX minus 2. All squared, OK. Plus, Ellen. X + 1 are squared, OK? And now using this definition, notice both of them have the same base so then we could rewrite it then as the product of both of those terms. The natural log of X minus 2 are squared multiplied by X + 1 R 2d. So we've shown that the left hand side is indeed equal to the right hand side, so our identity is true. Now, based on this given property as well, there are strict domain restrictions for logarithmic functions. Also, recall, OK. Let me put this in black here. We also know that for any form, for any log, so log based A of FF X or any log, then we know that F of X or function has to be greater than 0. So in this case, that means this term for the identity to be true must be greater than 0, which means it cannot be equal to zero. So if we think about it, right, we know then. That because both of these terms are squared, then the real value is going to be non-negative. The result will be non-negative, so it's squared so we don't have to worry about negative results for the luck to be true. The only thing we have to worry about then is that the points that are going to make each term equal to zero. So what is going to make each of our terms equal to 0? Well, not that. For X minus 2 are squad, when X is equal to 2, then X minus 22 will be 2 minus 22, which is gonna be equal to 02 or 0. Also, when X equals -1, OK, then the second term X + 1 squad will be -1 + 1 squad, which it also be equal to 0. Thus, when X equals 2 and X equals -1, our identity will not be true. Therefore, we can conclude that the equation is true. The equation is true. For all real values. OK. Excluding Uh, except when X equals -1 and X equals 2. I know if we were to write that in set notation, we're basically saying that X is an element of all real numbers except when it is -1 and 2. Therefore, these are the values of X that will make the equation true. If we look back at our answer choices, that tells us then that D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

b. ln(x + 1) + ln(x − 1) = ln(x² − 1), for all x.

Key Concepts

Properties of Logarithms

Domain of Logarithmic Functions

Counterexamples in Mathematics

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