Solve each equation. log￬9 x = 5/2

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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 four of X is equal to seven halves. We're given four answer choices. Option A, the set containing 128. Option B, the set containing 64 option C, the set containing eight and option D, the set containing 28. OK. So each of these answer choices is just one single solution. We're gonna start by rewriting what we were given. So we have a log base four of X is equal to seven halves. Now our X is inside of this log. OK. In order to solve for X, we need to get it outside. Now recall that we can relate a log equation to an exponential. Yeah. So if we have a log base B have something A is equal to Y, we can rewrite this as B to the exponent Y is equal to A, OK. So the base B of our log becomes the base B of our exponent. So let's take our log equation and rewrite it as an exponential equation so that we can solve for X, the base of our log is four. So the base of our exponent will be four. OK. The value of Y is gonna be the number on the opposite side of the log, maybe the opposite side of the equation. So that's seven halves. So we have four to the exponent, seven halves is equal to everything inside of the X K. That represents the A term which in this case is X and I think I said everything inside the X, what I meant was everything inside of the log. And in this case, that's X. OK. So now we have X is equal to four to the exponent seven halfs. OK. How can we simplify this? We recall if we have some number, say A to the exponent M multiplied by N, we can break up this exponent and write it as a to the exponent M to the exponent. So seven halves is equal to seven multiplied by one half. So we can break that exponent up and write this as X is equal to four to the exponent, one half all to the exponent set. Now forward to the exponent one half, recall that we can write exponents of one divided by something as roots. So this is equivalent to the square root of four. We have X is equal to the square root of four, up to the exponent seven. We know that the square root of four is equal to two. We have two to the exponent seven. And we have to be a little bit careful when we take the square root, we're gonna get both the positive and negative root. However, in this case, in our X is inside of the log, OK. In our original equation, that means we can't have a negative X value. OK. If we have a negative X value, we can't take the log of that value. OK? It doesn't exist. And so we're looking for only positive X values here. And so we're gonna take just the positive root. So you have two to the exponent seven now, which is equal to 100 and 28. And that's the final answer we were looking for. We found that the solution to this log equation is X is equal to 128 which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Solve each equation. log￬9 x = 5/2

Key Concepts

Logarithms

Exponential Form

Properties of Exponents

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