Solve each equation. x = log￬3 1/81

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Hey, everyone in this problem, we're asked to solve the following logarithmic equation. The equation we're given is X is equal to log base 11 of one divided by 1331. We're giving four answer choices. All four of them are sets containing a single value. Option A, that value is a negative three for option B, it's three for option C, it's two and for option D, it's negative two. We're gonna start by rewriting what we were given. We have X is equal to log of one divided by 1330 vote. Now, there are a couple of different ways we could approach this problem. The first one would be to leave the equation as it is and simplify this one divided by 1331 inside of our log. OK? Until we get things like exponents that we can use log rules to bring out and that sort of thing. The second way to approach this problem. And then what I'm gonna do is to rewrite this as an exponential equation. So if we have log base B of A is equal to some value Y, we can rewrite this as B to the exponent Y is equal to A. So the base of a log becomes a base of the expo in our base, the base of the log, the sorry, the base of the log is 11. So we have 11, the exponent is Y which is the value that's on the opposite side of the equal sign of the lock. In our case, that's X. So we have 11 to the exponent X is equal to A which is everything inside the lock one divided by 1331. So we've rewritten this as an exponential equation. And again, you could have done this a different way. Both of them are about the same length and they will both get you the same correct answer. Now we have this X and the X, if we can write the right hand side of our equation to be 11 to the exponent something, then we can compare the exponents. So let's try to do that. Now, 1331 this is actually 11 cubed. So we can write 11 to the exponent X is equal to one divided by 11. Cute. Now, this exponent is in the denominator, we wanna move it to the numerator. Recall that if we have one divided by something, say X to the exponent M, we can write this as X to the exponent negative M. OK. So we can move the base and the exponent to the numerator, we just have to switch that exponent to be negative. And so we get that 11 to the exponent X is equal to 11 to the exponent negative three. OK? Now we have 11 to some exponent. It's equal to 11 to some exponent. Those exponents must be equal in order for this equation to be true. OK? That means that X must be equal to negative three. So we have our solution here. X is equal to negative three which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Solve each equation. x = log￬3 1/81

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Key Concepts

Logarithms

Change of Base Formula

Exponential Equations