Solve each exponential equation in Exercises 1–22 by expressing e...

Question

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 8^(x+3)=16^(x−1)

Accepted Answer

everyone. Welcome back in this problem, we are asked to solve exponential equation by converting the terms into the same basis. We're given the equation here 27 to the X plus four Is equal to 81 to the X -5. Alright, and again, we're given a little bit of a hint that we want to first convert the terms into the same basis. Okay. All right. So looking at 27, we know we can rate this three to the power of three. And now if we want to write in terms of the same basis, let's try to write 81 in terms of The base three. Well, it turns out we can write three to the four. Okay, that will give us 81 and we're still left with the x minus spot. The experiment there. Okay, so we can write this as three to the exponent three all times X plus four is equal to three to the exponent four. All times x minus five. Alright, so now we have, the left hand side is equal to the right hand side, we have the same base on both sides. So in order for the expression to be equal, while the exponents must be equal. Okay, so we can set the exponents equal three times X plus four equals four times X minus five, expanding out three X plus 12. Remember that? The term in front needs to distribute to both terms inside the brackets on the right hand side. Four X minus will move the X terms to the right. So we're gonna have 12 left on the on the left hand side. We're gonna subtract three X from both sides. We'll get X minus 20 Okay? And finally moving that 20 over 12 plus 20 is going to be 32 so we get X is equal to 32. Okay? And that's going to correspond with solution C. Alright. I hope that helps. Thanks for watching everyone.

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 8^(x+3)=16^(x−1)

Key Concepts

Exponential Equations

Base Conversion

Equating Exponents

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