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. log(3x−3)=log(x+1)+log 4

Accepted Answer

Hey everyone welcome back in this video. We're asked to solve a logarithmic equation. Okay We're asked to reject any X. Values that make the original function undefined. And to calculate our solution to three decimal places. Okay? And the equation we're given is the natural log of seven X minus seven is equal to the natural log of X. Plus 12 plus the natural log of eight. Okay. So rewriting and I'm just going to refer to the natural log as lawn short forms lawn of seven X minus seven equals lawn of X plus plus lawn of eight. All right so we have a question like this. What we want to do is we want to simplify so we can get one single log rhythm on the left, one single log rhythm on the right. And that way we can compare the two. Okay so let's recall our log rhythm rules. We have one that says if we have a log of base B. of two things say eight times K. Well we can write this as a log base B. Of a. Plus log base B of K. Okay? So if they're multiplied within the log we can separate them into two separate logarithms each with the same base. Okay? So in this case we have log we have lawns we have log base E. Okay we have two terms so now we can create one term with them by multiplying. So on the left hand side we're still gonna have lawn of seven X minus seven. On the right hand side. We're gonna have lawn and then we're gonna have X plus all times eight. Okay So now we've simplified we have one single log rhythm on the left. Seven X minus seven on the right lawn and expanding and distributing that eight needs to go to both terms. We get eight X plus 96. Okay now we have the natural log of something on the left is equal to the natural log of something on the right. That means that there's some things have to be equal. Okay so we can compare the seven X minus seven and the A. X. Plus 96. Okay They must be equal. Now solving we're gonna move the excess to the right hand side and the numbers to the left. So moving the seven X. To the right we're going to get minus seven is equal to X plus 96. Okay and moving the 96 over to be with the -7 -7 -96 is going to give us -103. Okay now this is the answer we get but we need to check that it fits. Okay So what we want when we're looking at the domain of these functions with the domain Of log of seven X -7. Okay we need whatever is in the natural log to be bigger than zero. Okay so we need seven X -7 to be bigger than zero. That means that seven X. Has to be bigger than seven And divided by seven. That means X needs to be bigger than one. Okay. Well the answer he found is X equals minus one oh three. Okay, that's less than one. So that's not in the domain so that's gonna give us undefined function. Okay? So X equals minus 103 is not a solution. Okay. And so in this case we actually end up with no solutions. Okay? We found one potential solution but it doesn't fit within the domain of the original function. So it's not a solution. So d we have no solutions. Thanks everyone for watching this video. I hope it helped you.

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