Solve each equation. Give solutions in exact form. See Example...

Question

Solve each equation. Give solutions in exact form. See Examples 5–9. log(9x + 5) = 3 + log(x + 2)

Accepted Answer

Hello, everybody. I have everything. 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 log of 12 plus five X equal to two plus log of X plus seven. Now the answers they provide us are a negative 688 divided by 95 B 688 divided by 95 C 32 D no solution. Now, when looking at this equation, the first thing I can think of is putting similar terms on the same side of the equation. So I see I have a log on the left hand side and another log on the right hand side. So let's put them on the same side to have them together. So I'll move the log of X plus seven from the right hand side to the left hand side. So we have log of 12 plus five X. And now, since I moved the other log over, it'll be minus log of X plus seven and that'll be equal to two. Now, since I have two logs with the same base subtracting each other. On the left hand side, I can do a recall on different formulas and rules that we have for algorithms. In this case, the quo rule. So I'll write a general formula for that. Let's say we have log of base B of A minus log of base B of C that will be equal to log of base B of A divided by C. So basically, that's just telling us if you have two logs of the same base in this case, B and let's say you have the term A and the term C in each one and they're subtracting each other, they can be combined to be one log of the same base of the terms A and C dividing A divided by C. So let's just compare that to the equation that we have. So we have log and they don't specify any specific base. Therefore, you can assume that the base is 10. When they only write log with no other base, the base will be 10. So that will be the 10 will be the B in our recall. The term 12 plus five X will be the A in a recall and the term X plus seven will be the C in our recall. So using that information, let's just rewrite the equation that we have and it'll be log of 12 plus five X and all that divided by X plus seven. And that'll be equal to two. Now, if you just look at this from a point of view of you have a log equal to a number, it's just a simple log equal to a number. Well, that can be rewritten as an exponential equation. And that will lead me to do another recall of how to write logs and logarithms as exponential equations. Again, I'll write a general formula for that. Let's say you have log of base B of A equal to C that can be rewritten as B to the power C equal to A. So that's just a way to change from a logarithmic equation to an exponential equation, which is what we would do. In this case. Again, our base will be 10 and that will be the B in the recall, the entirety of 12 plus five X divided by X plus seven will be the A in the recall and the number two and the answer in this case will be the C in the recall. So using that information, let's rewrite once again. So we would have 10 squared equal to 12 plus five X divided by X plus seven. And we can do some simple and quick uh simplification where 10 squared will give us 100. And now we can just go ahead and start simplifying this equation. We have an X plus seven on the right hand side with which is dividing the 12 plus five X. So we can move that over to the left hand side and multiply it by the 100. So we'll have 100 multiplied by X plus seven, equal to 12 plus five X. Now, we can distribute the 100 over. So we'd have 100 multiplied by X and then multiplied by positive seven. So it would have 100 X plus seven plus 700. I'm sorry, equal to 12 plus five X still. So now it's just again, a matter of putting similar terms on the same side of the equation. So I'll have 100 X on the left hand side. And then I'll move the five X over from the right hand side to the left hand side. So it'll be 100 X minus five X now and then I'll move the 700 from the left hand side to the right hand side. So we'll have 12 on the right hand side or 100 X minus five X equal to 12 minus 700. Now, which will be 95 X equal to negative 688. And now to get the X alone will multiply will divide each side by which will be equal to negative divided by 95 which will be our answer for X. And the first instant would be to go to answer choice a and circle that because it matches what we have. However, before we can do that, we have to check if this X would work in this scenario. So let's say you have the negative 688 divided by 95. And let's say you plug that into the equation that we have and, and let's say you have log of 12 plus five X. If you multiply it five by the X that we have, we'll get a negative term inside of that log. And as we know, you cannot take the log of a negative number that would be inconclusive. Therefore, that X would not work in this case. So we have to eliminate it as a choice and we are left with no other answer except no solution for this case. So I hope that was helpful. And until next time.

Solve each equation. Give solutions in exact form. See Examples 5–9. log(9x + 5) = 3 + log(x + 2)

Key Concepts

Properties of Logarithms

Solving Logarithmic Equations

Domain Restrictions of Logarithmic Functions

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