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. (1/2)^x = 5

Accepted Answer

Hey, everyone. Here we are asked to solve the following equation for X and round the answer to the nearest 1000. Here we are given the equation, the quantity of 1/13 raised to the power of X is equal to 32. Here we have four answer choice options, answer a negative 0.952, answer B negative 2.745, answer C negative 1.351 and answer D negative 3.941. Looking at our given equation here, we have the variable X as an exponent of 1/13. And so we need to get X to be outside of this exponent. And so to do this, we need to manipulate our given equation by first taking the natural log of both sides. So doing this, we have the natural log of the quantity of 1/ raised to the power of X is equal to the natural log of 32. So now here we still have X as an exponent. And so we need to apply the logarithmic property which states that the natural log of X raised to the power of A is equivalent to a times the natural log of X. So now we take this property and apply to the left side of our equation where again, we have X as the exponent of 1/13. And so applying this property, we can rewrite our equation here as X times the natural log of 1/13 is equal to the natural log of 32. So now the next step is to just get X alone. And so all we have to do here is divide both sides by the natural log of 1/13. And so rewriting our equation, we see we now have X alone and that X is equal to the natural log of 32 divided by the natural log of 1/13. So now the last step here is to plug in this expression we found for X into our calculator and approximate the answer to the nearest thousands. So we need to plug in the expression the natural log of divided by the natural log of 1/13 into our calculators. And we see that our approximate solution for X is negative 1.351. So with this final answer here, we are left with answer choice C where again our solution for X is negative 1.351. 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. (1/2)^x = 5

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