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

Question

Solve each equation. Give solutions in exact form. See Examples 5–9. ln(5 + 4x) - ln(3 + x) = ln 3

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 the natural log of seven plus nine X minus the natural log of five plus X and that will be equal to the natural log of eight. Now, the answers that they provide us are A which is two answers eight and zero B 14 C 33 D no solution. Now, when I look at our equation, there's multiple approaches that we can take here and I'll talk about those by doing a simple recall of different rules that we have for algorithms. So what we have right now is two natural logs subtracting each other. So we can talk about the quotient rule and I'll write a general formula for that. Let's say you have log of base B minus log of base B of C, that'll be equal to log of base B of A divided by C. So what that's telling us is if you have two logs of the same base in this case, base B and one being of A and the other one of C and they're subtracting each other, you can have one log of that same base of A divided by C. So comparing that to the, in the equation that we have, we have two natural logs, which means they have the same base. A natural log is just log of base E. So that base will be the B in a recall seven plus nine X term will be the A in our recall. And the five plus X term will be the C in a recall. So another approach that we can take to this is, let's say you move over the minus L N or minus natural log of five plus X, you move it over from the left hand side of the equation to the right hand side. So you add it to the right hand side of the equation that will be rewritten as natural log of seven plus nine X equal to the natural log of eight plus the natural log of five plus X. So now we have instead of having two log, two natural logs subtracting each other. Now we have them adding each other and we can talk about the product rule. So let's say I'll write a general formula again. Let's say you have log of base B of A plus log of base B of C that can be rewritten as, or it will be equal to log of base B of a multiplied by C. So this is the same concept as the quotient rule except now the terms A and C in this case will be multiplying each other not dividing. So looking at the equation that we have now, and this is the formula and the, the method that I'm gonna be using simply because I prefer using the product rule over the quo rule just to avoid having division and certain fractions in our work. Uh It'll make it just a little bit easier. So, but again, you can mention um or you can use either method. So moving on with the product rule, let's compare it to what we have and see what we can do. Again, we have natural logs. So our base E will be the B in the recall. The term eight will be the A in a recall and the term five plus X will be the scene in our recall. So using that information, let's rewrite what we have and it'll be natural log of seven plus nine X equal two. And now using the product tool, we have the natural log of eight multiplying five plus X and to do some simplification, what we can do is distribute that eight multiplying the five plus X. So we'll have natural log of seven plus nine X which stays the same equal to the natural log of. And as I distribute, we would have eight multiplying the five which would be and then eight multiplying a positive X. So we'll have 40 plus eight X. No, we have the left hand side of the equation and the right hand side of the equation being a natural log and no other term. So because of this, the equation is telling us that the natural log of seven plus nine X is equal to the natural log of 40 plus eight X. Therefore, seven plus nine X must be equal to 40 plus eight X since their natural logs will be the same. So to simplify this, we can drop the natural logs and just have seven plus nine X equal to 40 plus eight X. And now it's just a matter of getting similar terms on the same side of the equation and simplifying. So we'll have, we'll move the eight X from the right hand side of the equation to the left hand side. So we'll have nine X minus eight X and we'll move the seven from the left hand side of the equation to the right hand side. So we'll have nine X minus eight X equals 40 minus seven. And now it's just a matter of simplifying. So nine X minus eight X is just X equal to minus seven, which is 33. And that will be our answer for this question or letter C. In this case, I hope that was helpful. And until next time

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Solve each equation. Give solutions in exact form. See Examples 5–9. ln(5 + 4x) - ln(3 + x) = ln 3

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