Solve each equation. log￬x 25 = -2

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Hey, everyone in this problem, we're asked to solve the following logarithmic equation for X. The equation we're given is log base X of 81 is equal to negative four. We're given four answer choices. All four of them are sets containing a single value. Option A is the value three, option B one quarter, option C one third and option D four. Now we're gonna start by rewriting the expression we were given or the equation we were given log base X of 81 is equal to negative four. Now this is in the base of our log. OK. So in order to solve for this equation, we need to rewrite this so that we can get that base of that log out. Now let's recall, we have A that allows us to write logarithmic equations as exponential equations. So if we have log space B have something, say A is equal to Y, we can rewrite this as B to the exponent Y is equal to A. Yeah. So the base B of A log becomes the base B of our exponent. Now, in our case, the base of our log is X and so X is gonna become the base of our X. The exponent is the value on the opposite side of the equal sign to our look. OK? So that's that Y value and in our case, that's negative four. So we have X to the exponent negative four and this is gonna equal what's inside of the log that a value which in our case is 81. So we have X to the exponent negative four is equal to 81. Now recall if we have a, to the exponent M, we raise it to the exponent and we can simplify that exponent and write it as a to the exponent M multiplied by N. So in order to solve for X, what we want is an exponent of what? OK. Right now we have an exponent of negative four. We can raise the left-hand side to another exponent. OK? We want that exponent multiplied by negative four to give us one. So if we take X to the exponent negative four and we raise it to the exponent negative one quarter, then when we use the power rule to simplify these exponents, we have negative four multiplied by negative one divided by four and we get just one. Now if we do that on the left hand side, we have to do it to the right hand side as well. So we have X to the exponent negative four to the exponent negative a quarter is equal to 81. To the exponent negative a quarter. Now simplifying on the left hand side, we just talked about this, we multiply the exponents together, we're gonna get X to the exponent one, which is what we wanted. On the right hand side, we have a negative exponent. If we have a to the exponent negative M recall that we can write this as one divided by A to the exponent M OK. So we can move the base and the exponent to the denominator and that exponent is going to become positive. So 81 to the exponent negative a quarter can be written as one divided by 81 to the exponent positive a quarter. Mhm All right, on the left hand side, we have just X on the right hand side, one divided by 81 to the exponent one quarter. Now recall if we have a to the exponent one divided by N, this is equivalent to the nth root of A OK. So this exponent of a quarter is actually equivalent to 1/4 root. We get that this is equal to one divided by the fourth root of 81. Now the fourth route, what we want to do is find a value where we multiply it four times we get 81 and that value is three. OK. So we get that X is equal to one divided by three. And that is the final solution we were looking for. OK. This corresponds with answer Choice. B thanks everyone for watching. I hope this video helped see you in the next one.

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Solve each equation. log￬x 25 = -2