Solve each equation. Give solutions in exact form. See Examples 5...

Question

Solve each equation. Give solutions in exact form. See Examples 5–9. log x + log(x - 21) = log 100

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're gonna be looking at this math question that's asking us to solve the logarithmic equation for X and provide an exact solution where our equation is log base six of X plus log base six of X minus nine that will be equal to log base six of 36. Now the answers that they provide us are a negative three comma 12 B 12 C three and D negative 12 comma three. Now, by looking at our equation, the first thing I can notice is that we have two logs on the same side with the same base and they're adding each other. So I can do some recall of different formulas and different rules that we have for algorithms. This one specifically will be the product rule. So I'm just going to write a general formula for that. Let's say we have log of base B of A plus log base B of C that will be equal to and can be rewritten as log base B of A multiplied by C. So comparing that to our, to our equation, we can see that the base six will be the base B in our recall. X will be A in our recall and X minus nine will be the C in our recall. So using that information, let's rewrite what we have. So we will have log base six of X multiplied by X minus nine and that will still be equal to log base six of 36. Now, we can do some distribution with the X multiplying the X minus nine. So let's do that. We'll have log base six of X squared minus nine X and that will be equal to log base six of 36. Now, both sides have a log of base six and no other term, the left side has log base six of X squared minus nine X and the right side that and that is equal to log six of 36. Therefore, because they're both log of base six, the inside term they must be equal to each other. So the X squared minus nine X must be equal to 36. So we can drop the log may six and just write that X squared minus nine X will be equal to 36. Now, we can move the 36 over to have X squared minus nine X minus 36 will be equal to zero. And with this information, we can do some factoring. So let's do that. So we'll have XX minus 12 multiplied by X plus three and that will be equal to zero. And we just factored the X squared minus nine X minus 36. So now after doing that factory, that'll tell us that X minus 12 will be equal to zero and X plus three will be equal to zero. So one X will be positive and the other X will be negative three. Now, before picking an answer choice, we must look at the X values that we got and see if they would work in the original equation. When looking at X of negative three, we can right away see that that will not work when plugging that back in right into the first term log base six of X. If we have log base six of negative three, that'll be inconclusive because we cannot have log of a negative number. Therefore, we can eliminate X equal to negative three and we will be left with X equal 12 because we are able to plug that into the equation that we have with no problem. Therefore, our answer for this question will be answer choice B or 12. I hope that was helpful. And until next time.

Solve each equation. Give solutions in exact form. See Examples 5–9. log x + log(x - 21) = log 100

Key Concepts

Properties of Logarithms

Solving Logarithmic Equations

Domain Restrictions of Logarithmic Functions

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