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^4x+5e^2x−24=0

Accepted Answer

Hey everyone here we are asked to solve the exponential equation E. To the power of four X plus six, E. To the power of two, X minus 55 is equal to zero and were asked to give our answers in terms of natural logarithms and its decimal approximation. So to start we see that we have a rather complicated equation to factor. So we want to work to simplify it. So doing this we need to recall the power to a power exponential rule allowing us to rewrite this equation as E. To the power of X to the power of four plus six times E. To the power of X to the power of two minus 55 is equal to zero. And now to simplify this here we can let you be equal to E. To the power of X. So we can rewrite the equation as you to the power of four plus six times U. To the power of two minus 55 is equal to zero. And now we can just factor this equation to the quantity of U squared minus five times the quantity of U squared plus 11 is equal to zero. Now we can immediately rule out this U squared plus 11 because it gives us U. Is equal to the square root of negative 11 which we don't want because we can't take the negative the square root of a negative value and now we can just focus on this U squared minus five. Instead it equal to zero. So finding the value of U, we first add five to both sides giving us U squared is equal to five. Now to get rid of the square, we take the square root of both sides giving us U. Is equal to plus or minus the square root of five. And now here we want to revert this you back to E. To the power of X. So we really have E. To the power of X equal to the positive square root of five. And we also have E. To the power of X is equal to the negative square root of five. And now we just want to convert both equations to the natural logarithms form. So to do this we take the natural log of both sides. So beginning with the positive square root of five equation, we have the natural log of E. To the power of X is equal to the natural log of five to the power of one half which is again the square root of five. So doing this, we can now take the powers and bring them to the outside because we're taking the natural log. So we have X times the natural log of E is equal to one half times the natural log of five. And now with this we need to recall that the natural log of E is equal to one. So we really have the X is equal to one half natural log of five. And now we solve this equation. So we can move on to the negative square root of five equation. And repeating the process, we take the natural log of both sides. So we have the natural log of each. The power of X is equal to the natural log of negative five to the power of one half. Now we can't take the natural log of a negative value. So this expression here because of this negative value gives us no solution. And so with this we see that our first equation with the positive square root of five is going to be our final answer. So we see that X is equal to the natural log of five divided by two which is it has a decimal approximation of 0.805. And this leaves us with answer choice. A 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^4x+5e^2x−24=0

Key Concepts

Exponential Equations

Natural and Common Logarithms

Calculator Use for Approximations

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