Solve each equation. log￬1/3 (x+6) = -2

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Accepted Answer

Hey, everyone in this problem, we're asked to solve the following logarithmic equation for X. The equation we're given is log base 16 of X plus 185 is equal to negative three. We're given four answer choices. Option A, the set containing negative 13, option B, the set containing 31 option C, the set containing 27. And option D there is no solution. Let's start by writing what we were given. So we have log base 1/6 of X plus 185 is equal to negative three. Never call. If we have log base B of A is equal to Y, we can turn this into an exponential equation. We can write this as B to the exponent Y is equal to A. OK? So let's go ahead and do that. We'll write it as an exponential equation. Maybe it'll be easier to deal with in that way. So the base of our log becomes the base of a phone. OK? So we have 1/6 is the base of our log that becomes the base of a phone. So we get 1/6 the Y value in our expression is going to be negative three. That's what's on the opposite side of the log that becomes our exponent. So we have 16 to the exponent negative three. And that's equal to everything inside of our log, which is X plus 185. Now, let's expand on the left hand side, we have one divided by six all to the exponent negative three. OK. When we have a division raised to an exponent, we need that exponent to apply to both the numerator and denominator. So applying it to the numerator, one to the exponent negative three just gives us one. OK. And so we can write the left hand side as one divided by six to the exponent negative three. On the right hand side, we still have X plus 185. Now we have a negative exponent in the denominator. How can we make this a positive exponent overall? If we have one divided by say A to the exponent negative N, you can write this as a to the exponent positive M so we can move the base and the exponent from the denominator up to the numerator and that negative exponent will become positive. So on the left hand side, what we actually have is six cubed and that is equal to X plus 185. Now six cube, remember six multiplied by six, multiplied by six. This is gonna give us 216 is equal to X plus 185. And to isolate for X, we're gonna subtract 185 from both sides. We get that X is equal to 31. All right. Now, if we look at our original equation, we want to double check that this is indeed a solution. If we were to put X equals 31 into our original equation, we're taking the log of something that's positive. So there's no issues there with the domain or with any restrictions. And so X equals 31 is a solution. And we get that the correct answer is option B the set containing 31. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each equation. log￬1/3 (x+6) = -2

Key Concepts

Logarithmic Functions

Change of Base Formula

Solving Exponential Equations

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