Solve each exponential equation in Exercises 23–48. Express the s...

Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^2x−3e^x+2=0

Accepted Answer

Hey everyone here we are asked to solve the exponential equation into the power of two X minus 13. E. To the power of X plus 12 is equal to zero. And were asked to give our answers in terms of natural logarithms. So to start we want to manipulate our given equation so it is more simple two factor. So we first want to recall the exponential role of a power to a power, allowing us to to rewrite this equation as E. To the power of X to the power of two minus 13 times E. To the power of X plus 12 is equal to zero. And now from here we want to let you be equal to E. To the power of X. So we can simplify this equation allowing us to rewrite it as U. To the power of two minus 13 times. You plus 12 is equal to zero. And now we have a much easier equation to factor. So it factors into the quantity of U minus 12 times the quantity of U minus one is equal to zero. So now we can set both sides equal to zero to find our values for you. So we have U minus 12 is equal to zero, showing that U. Is equal to 12 and then we have U minus one is equal to zero. So we see that U. Is also equal to one. And now we want to revert our use back into the E. To the power of X. So doing this, we have E. To the power of X is equal to 12 and we also have E to the power of X is equal to one. And so now we want to convert these equations into natural logarithms. So to do this we take the natural log of both sides. And starting with E. To the power of X is equal to 12. We have the natural log of E. To the power of X is equal to the natural log of 12. And now we can rewrite the natural log of E to the power of X by taking out this X power to the outside of the natural log which gives us X times the natural log of E is equal to the natural log of 12. And now if we recall the natural log of E is just equal to one. So we have X is equal to the natural log of 12. And now for our second um equation we again take the natural log of both sides. So we have the natural log of E to the power of X is equal to the natural log of one. And now here we see the natural log of one is always equal to zero. And we can rewrite the natural log of E to the power of X as just X as we found earlier. So we have X is equal to zero. And so now we see we have two values of X. We have X is equal to the natural log of 12 and we have X is equal to zero, which gives us answer choice. D. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e^2x−3e^x+2=0

Key Concepts

Exponential Equations

Natural and Common Logarithms

Calculator Usage for Approximations

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