Solve each equation. In Exercises 11–34, give irrational solution...

Question

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4.
3(2)^(x-2) + 1 = 100

Accepted Answer

Hey, everyone. Here we are asked to solve the following equation for X and round the answer to the nearest thousands. Here we are given the equation eight times five raised to the power of X minus seven plus three is equal to 67. Here we have four answer choice options. Answer a 8.292, answer B 5.918, answer C 6.225 and answer D 7.155. So here to begin solving our given equation for X, we want to manipulate this equation so that we get five alone. And this is because five is raised to a power including the variable X, which is what we are trying to solve for. So we can start by simply subtracting three from both sides. In doing this, we have eight times five raised to the power of X minus seven is equal to 67 minus three, which gives us 64. Now continuing to get five alone, we need, need to get rid of the coefficient of eight. And we do this by dividing both sides by eight. And so we are left with five raised to the power of X minus seven is equal to eight. So now we see again, we have X in an exponent of five. And so to get X outside of this exponent, we need to take the natural log of both sides of the equation. So on the left side, we have the natural log of five raised to the power of X minus seven. And on the right side, we have the natural log of eight. So now looking again at the left side, we need to recall here that the natural log of X raised to the power of A is equivalent to a times the natural log of X. So now with this property of the natural log in mind, we can rewrite the left side of our equation so that we get X out of the exponent here. So doing this, we have the quantity of X minus seven times the natural log of five and this is still equal to the natural log of eight. So now we want to get rid of the natural log of five on the left side. So we need to divide both sides of our equation by the natural log of five. So so far, we now have X minus seven is equal to the natural log of eight divided by the natural log of five. And now we see here, the last step in getting X completely alone is to move this negative seven to the right side of our equation. And we can do this by just adding seven to both sides of our equation. And we see here, we are left with X is equal to the natural log of eight divided by the natural log of five plus seven. So now the last step here is to just use a calculator to approximate this answer here. Where again, in our calculator, we plug in the natural log of eight divided by the natural log of five plus seven and doing this and rounding to the nearest thousands, we get the value 8.292. So with this, we see our value for X from this given equation is 8.292. So with this value of X, we are left with answer choice. A 8.292. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. 3(2)^(x-2) + 1 = 100