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_2 (x^2 - 100) - log_2 (x + 10) = 1

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 of base four of X squared minus minus log of a four of X plus 15. And that'll be equal to one. Now, the answers that they provide us are a B 25 C comma 25 D 28. Now, looking at the equation that we have, I can see that I have two logs of the same base subtracting each other. So I can do a recall of some different rules that we have for logarithms. And in this case, it'll be the quotient rule. And I'll use a general formula. Let's say we have log of base B of A minus log of base B of C that'll be equal to, or it can be rewritten as log of base being of A divided by C. So using that recall ruling, comparing it to the equation that we have, our base four will be the base B in the recall, the term X squared minus 225 will be the A in a recall and the term X plus 15 will be the C in our recall. So using that information, let's rewrite the equation that we have. So we will now have log of base four of X squared minus 225 divided by X plus 15 and all of that equal to one. Now, we have a logarithm equal to a number and we can use another recall rule on how to turn a logarithmic equation into an exponential equation. So again, I'll write a general formula that we can use. Let's say we have log of base B of A equal to C that can be rewritten as B to the power C equal to A. So comparing that to the equation that we have, now our base four will be the B in our recall, the term X squared minus 225 divided by X plus 15 will be the A in our recall and the number one will be the scene. So let's rewrite this. After looking at the recall, we will have four to the first power equal to X squared minus 225 over X plus or divided by X plus 15. Now, we can do some simplification. We know that four to the first power is just gonna be equal to four. So we'll have four equals X squared minus 100 and 25 divided by X plus 15. Now, before we move on, we can do some simplification on our numerator, X squared minus 225 because X squared minus 225 has a very special quality and that is, it's a difference of squares. So we can rewrite it as X plus 15. So we have four equals X plus 15, multiplying X minus 15 and all of that divided by X plus 15. So now with, we have X plus 15 in the numerator and X plus 15 in the denominator, we can cancel those two terms out and will be left with four equal to X minus 15. So finally, we can bring the 15 over, add it to the four and we'll have that X is equal to and that will be our answer for this problem or letter A. So I hope that was helpful. And until next time.

Solve each equation. Give solutions in exact form. See Examples 5–9. log_2 (x^2 - 100) - log_2 (x + 10) = 1

Key Concepts

Properties of Logarithms

Exponential Equations

Domain Restrictions

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