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Question Video: Finding the Solution Set of Logarithmic Equations over the Set of Real Numbers Mathematics • 10th Grade

Find the solution set of log_๐‘ฅ (5) + log_๐‘ฅ (40) โˆ’ 2 log_๐‘ฅ (4) = 2 + log_๐‘ฅ (8) in โ„.

02:55

Video Transcript

Find the solution set of log base ๐‘ฅ of five plus log base ๐‘ฅ of 40 minus two times log base ๐‘ฅ of four is equal to two plus log base ๐‘ฅ of eight in all real numbers.

First, weโ€™ll copy down our equation. And though there are a few strategies we could use, letโ€™s just start by working from left to right. So we have log base ๐‘ of ๐‘ฅ plus log base ๐‘ of ๐‘ฆ, which will be equal to log base ๐‘ of ๐‘ฅ times ๐‘ฆ, which means we can combine our first two terms and call them log base ๐‘ฅ of five times 40, log base ๐‘ฅ of 200. And then weโ€™ll just bring everything else down.

Weโ€™re close to being able to combine log base ๐‘ฅ of 200 and two times log base ๐‘ฅ of four. And we can use the rule that ๐‘ times log base ๐‘ of ๐‘ฅ is equal to log base ๐‘ of ๐‘ฅ to the ๐‘ power, which means two times log base ๐‘ฅ of four is equal to log base ๐‘ฅ of four squared. And then weโ€™re subtracting two logs that have the same base. And we know when that is the case, we can divide each of their values. log base ๐‘ฅ of 200 minus log base ๐‘ฅ of four squared will be equal to log base ๐‘ฅ of 200 over 16.

It seems like thatโ€™s all we can do here on the left side of the equation. What we can do now is we can move this log base ๐‘ฅ of eight to the left side of the equation by subtracting log base ๐‘ฅ of eight from both sides. We now have log base ๐‘ฅ of 200 over 16 minus log base ๐‘ฅ of eight. And weโ€™ll use this subtraction rule again, which will be log base ๐‘ฅ of 200 over 16 divided by eight. We know that divided by eight is the same thing as multiplied by one-eighth. And if we do that simplification, we end up with log base ๐‘ฅ of 25 over 16 is equal to two.

And now we want to take this out of logarithm form and put it into exponent form. If we have log base ๐‘ of ๐‘ฅ equals ๐‘˜, then we can rewrite that as ๐‘ฅ equals ๐‘ to the ๐‘˜ power. So we have 25 over 16 is equal to ๐‘ฅ squared. By taking the square root of both sides of the equation, we get that ๐‘ฅ is equal to plus or minus five over four, since the square root of 25 is five and the square root of 16 is four. Since weโ€™re only looking for values that are in the set of all reals, we want to say that ๐‘ฅ cannot be negative five-fourths. We donโ€™t wanna deal with a negative base here, as this would yield an answer that is imaginary. And so ๐‘ฅ is only equal to five-fourths, which makes the solution set just five-fourths.

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