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^x = 7

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^x = 7

Accepted Answer

Hey, everyone here we are asked to solve the following equation for X and round the answer to the nearest thousands. Here, our giving equation is eight races to the power of X is equal to 21. Here we have four answers. Choice options. Answer a 1.464, answer B 2.987, answer C 4.73 and answer D 1.992. So recalling our given equation again, we have eight raised to the power of X is equal to 21. And here we see our X variable is an exponent of eight here. And so we want to get X to be outside of this exponent so that we can solve this equation for X. So our first step here is to just take the natural log of both sides of the equation. So doing that, we have the natural log of eight raised to the power of X is equal to the natural log of 21. Now to get X to be outside of this exponent here we have to recall the logarithmic property where we know that the natural log of X raised to the power of A is equivalent to a times the natural log of X X. So now we take this property and apply it to the left side of our equation so far. So now rewriting our equation, we have X times the natural log of eight is equal to the natural log of 21. So now the next step is to get X alone by dividing both sides of our equation by the natural log of eight. So now we see that X alone is equal to the expression, the natural log of 21 divided by the natural log of eight. So now the last step here is to plug in this expression, we found for X into a calculator to a approximate our answer here to the nearest thousands. So we need to plug into our calculator. The natural log of 21 divided by the natural log of eight. And in doing this, our solution for X rounded to the nearest 1000 is 1.464. So with this final value for X, we are left with answer choice. A 1.464. 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. 3^x = 7

Verified Solution

Key Concepts

Exponential Equations

Logarithms

Irrational Numbers