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. 3^2x+3^x−2=0

Accepted Answer

Hey everyone here we are asked to solve the exponential equation of five to the power of two X plus six times five to the power of x minus seven is equal to zero. And we are also asked to give our answer in terms of natural or common logarithms as well as its decimal approximation. So here looking at our equation we want to simplify it as much as possible so it's easier to factor. So doing this. We first want to recall the exponential rule for a power to a power allowing us to rewrite this first term as five to the power of X to the power of two. And then we just have plus six times five to the power of X minus seven is equal to zero. And now from here we can simplify this equation by letting you be equal to five to the power of X. And now we can rewrite the equation as U to the power of two plus six times u minus seven is equal to zero. And now we can simply factor this equation. So we have the quantity of u minus one Tennessee quantity of U plus seven is equal to zero. And now from here we can set both sets of parentheses equal to zero. So we first have u minus one equals zero adding one to both sides. We see that U is equal to one, then we have you plus seven is equal to zero subtracting seven from both sides. We see that you is also equal to negative seven and now we want to revert the U back to its original value. So we actually have five to the power of X is equal to one. And we also have five to the power of X is equal to negative seven. And now we can convert these equations to the natural logs by taking the natural log of both sides. So um for the first equation of five to the power of X is equal to one. We'll take the natural log of both sides. Giving us the natural log of five to the power of X is equal to the natural log of one. And so we can recall that the natural log of one is simply zero. And we also have the natural log of five to the power of X. Which we can simplify by bringing X to the outside and having X times the natural log of five. But we can see that X will just be equal to zero. And now with this we can move on to the second equation and take the natural log of both sides. So we have the natural log of five to the power of X is equal to the natural log of negative seven. Well if we recall we can't take the natural log of a negative value. So this second equation here brings no solution. And so we are left with a final answer where X is just equal to zero which is 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. 3^2x+3^x−2=0

Key Concepts

Exponential Equations

Logarithms

Calculator Use for Approximations

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