Solve each logarithmic equation in Exercises 49–92. Be sure to re...

Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log2(x−6)+log2(x−4)−log2 x=2

Accepted Answer

Hey everyone welcome back in this problem. We are asked to solve the following logarithmic equation, reject any X value that makes the original function undefined and to use a calculator to approximate solution to three decimal places. Okay and the equation we're given is log base four of x minus three plus log base four of x minus nine minus log base four X is equal to two. Okay we have something like this. What we want to do is try to combine it into a single log rhythm. Okay We have the same base in each of the log terms we have so let's try to combine it into one single log. Alright so let's recall our log roll. So let's write out, we have log base four of x minus three plus log base four of x minus nine minus log base four of x equals two. Okay now we're going to deal with the first two terms first. So when we have addition between logs with similar bases, let's recall the rule, we have log base B of something. We call a plus log base B of something we call K. Okay well that's going to equal log base B of a times K. Okay so we can bring those together and have them multiply. So we're gonna apply that to the first two terms. We're gonna get log base four of x minus three all times X minus nine. Okay we still have minus log base four of X is equal to two. Okay. Alright so I'm just going to highlight here in red. We used this rule to deal with those two terms and get them into one combined log. Okay now in the next line we're going to try to deal with this second log and combine it with the others similarly to the addition rule, we have the subtraction rule. So log base B of a minus log base B F K. Is equal to log base B of A. Divided by K. Okay, so this is our next rule, This attraction rule, it's in blue. Okay and that's gonna let us combine These two. Okay, so now we're gonna have log base four of X -3 times X -9. All divided by X. Okay, that's all within one log on the right hand side, we have our two. Okay, so we've used that blue rule to combine that second log that subtracted into one log. Alright, so now we have our combined log rhythm and we're going to solve one more log world we're going to use here. Remember that log base B. Of A equals K. Can be written rewritten as B. To the K equals a. Okay, so be the base of the log rhythm becomes the base of this exponent? Exponent. Okay. Alright. So in terms of our problem when we're looking at this rule, Our base is four. So we have four to the exponents of the thing that we're equal to which is to And that's going to equal everything within the log. So that's going to be X -3 times X -9. All over X. Okay. Alright, so we've used our log rules and now we get an equation that looks familiar. Okay, this is a quadratic equation, we know how to solve those. So now it's just a matter of solving so multiplying both sides by X. We get 16 x on the left, X -3 times x -9 on the right. Okay, expanding 16 x is equal to x squared. Okay, we expand both terms. We get -12 x plus 27. Moving the 16 X to the right hand side. So we have a quadratic equals zero. We get zero equals X squared minus 28 X plus 27. And we can factor this one. So we factor X - An X -1. Okay. We're just gonna scroll down just a bit. So we have some more room here. Alright, so that's gonna give us X is equal to one Or X is equal to 27. Okay, so we have potential X values now. We need to check them to make sure That the original function is defined there. Okay, so when we're looking at our original function, we have three terms that we need to consider, we have the log of X -3 term. Base four. We have the log base four of x minus nine term and we have the log base four of extra. Okay. We know when we have log base four of X. That we need X bigger than zero. Okay In this case both of our quantities are bigger than zero. So that's good. With our X -9 term. We're going to need X -9 bigger than zero. Okay. We can't take the logarithms of something less than zero. Um so we need them bigger than zero. This means we need X bigger than nine. Alright well 27 is bigger than nine but one is not. Okay so we're going to cross out one that is not going to be a solution. Okay so X equals one. Sad face doesn't work. X equals 27. That works alright. And finally for the last one X minus three is bigger than zero. So X. We need to be bigger than it's not bigger than zero but bigger than three. Okay well that's already covered with X. Is bigger than nine. If X is bigger than nine then it's also bigger than three. Okay so again X equals one fails this but X equals 27 is good. Okay so our answer are one solution is going to be X equals 27. That's answer. B. I hope this video helped. Thanks for watching

Key Concepts

Logarithmic Properties

Domain of Logarithmic Functions

Solving Logarithmic Equations

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