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 1–4.0.05(1.15)^x = 5

Accepted Answer

Hey, everyone. Here we are asked to solve the following equation for X and round the answer to the nearest thousands here. The given equation is 0.1 times 2.55 raised to the power of X is equal to 12. Here we have four answer choice options. Answer a 4.864, answer B 7.291 answer C 5.114. And answer choice D 6. here with our given equation again, we have 0.1 times 2.55 raised to the power of X is equal to 12. Well, we want to manipulate this equation so that we first get 2. alone since it contains a power of X, which is what we are trying to solve for. So first, we need to get rid of the coefficient here which is 0.1. So we just need to divide both sides of the equation by 0.1. And in doing this, we have 2.55 raised to the power of X is equal to 12 divided by 0.1 which simplifies to 120. So again, so far, we have 2.55 raised to the power of X is equal to 120. So now we see that again, we have X as a power when we want X to be out of this exponent here. So to do this, we just need to take the natural log of both sides of our equation. And in doing this, we again, on the left side, we have the natural log of 2.55 races to the power of X. And then on the right side, we end up with the natural log of 120. Well, now, here we still have X as an exponent. But if we recall the property where the natural log of X raised to the power of A is equivalent to a times the natural log of X, we can rewrite this expression on the left side of our equation so far. And then we see that we will have X times the natural log of 2.55 is equal to the natural log of 120. So now we want to continue to get X alone. So we just need to divide both sides by the natural log of 2.55. And so now rewriting our equation, we see that X alone is equal to the natural log of 120 divided by the natural log of 2.55. So now from here, the last step is to use a calculator to a approximate the answer to the nearest thousands. So in our calculator, we just need to plug in the natural log of 120 divided by the natural log of 2.55. And we get the approximate answer of 5.114. So we see here that our solution, our value for X is 5.114, which leaves us with answer choice C where again, our value is 5.114. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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 1–4.0.05(1.15)^x = 5

Key Concepts

Exponential Equations

Logarithms

Rounding and Decimal Approximation

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