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.
0.8^x = 4

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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 0.6 raised to the power of X is equal to 18. Here we have four answer choice options, answer a negative 2.449, answer B negative 5.658, answer C negative 3.112 and answer D negative 4.938. So looking at our given equation again, we have 0.6 raised to the power of X is equal to 18. Well, here we see our X term is in the exponent of 0.6. So we want to manipulate this equation so that we get X outside of this exponent. And this means we just need to take the natural log of both sides of our equation. So we have the natural log of 0.6 raised to the power of X is equal to the natural log of 18. Now to get our X variable outside of this exponent here, we need to recall the logarithmic function 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 again, applying this property to the left side of our equation. So far, we can rewrite this equation as X times the natural log of 0.6 is equal to the natural log of 18. So now we want to get our X variable alone since again, we are solving this equation for X. And so we just need to divide both sides here by the natural log of 0.6. So again, we divide both the left and right hand side by the natural log of 0.6. And now we see that X alone is equivalent to the expression where we have the natural log of 18 divided by the natural log of 0.6. So now the last step here is to plug in this expression we found for X into our calculator to approximate the answer to the nearest thousands. So plugging in the natural log of 18 divided by the natural log of 0.6, we get the approximate solution where X is equal to negative 5.658. So now with this approximated answer for X, we are left with answer choice B where again we have negative 5.658. 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. 0.8^x = 4

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Key Concepts

Exponential Equations

Logarithms

Rounding and Decimal Approximation