Solve the following exponential equation. Express irrational solutions in exact form.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice,
A. The solution set is
(Simplify your answer. Use a comma to separate answers as needed. Use integers or fractions for any
numbers in the expression. Type an exact answer, using radicals as needed.)
B. There is no solution.

Ask by Harrington Gough. in the United States

Jan 21,2025

Question

Solve the following exponential equation. Express irrational solutions in exact form.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice,
A. The solution set is
(Simplify your answer. Use a comma to separate answers as needed. Use integers or fractions for any
numbers in the expression. Type an exact answer, using radicals as needed.)
B. There is no solution.

Ask by Harrington Gough. in the United States

Jan 21,2025

UpStudy AI · Accepted Answer

Solve the equation $$ 100^{x}+10^{x+1}-8=0 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$100^{x}+10^{x+1}-8=0$$
- step1: Rewrite the expression:
  $$100^{x}+10	imes 10^{x}-8=0$$
- step2: Rewrite the expression:
  $$10^{2x}+10	imes 10^{x}-8=0$$
- step3: Use substitution:
  $$t^{2}+10t-8=0$$
- step4: Solve using the quadratic formula:
  $$t=\frac{-10\pm \sqrt{10^{2}-4\left(-8ight)}}{2}$$
- step5: Simplify the expression:
  $$t=\frac{-10\pm \sqrt{132}}{2}$$
- step6: Simplify the expression:
  $$t=\frac{-10\pm 2\sqrt{33}}{2}$$
- step7: Separate into possible cases:
  $$\begin{align}&t=\frac{-10+2\sqrt{33}}{2}\&t=\frac{-10-2\sqrt{33}}{2}\end{align}$$
- step8: Simplify the expression:
  $$\begin{align}&t=-5+\sqrt{33}\&t=\frac{-10-2\sqrt{33}}{2}\end{align}$$
- step9: Simplify the expression:
  $$\begin{align}&t=-5+\sqrt{33}\&t=-5-\sqrt{33}\end{align}$$
- step10: Substitute back:
  $$\begin{align}&10^{x}=-5+\sqrt{33}\&10^{x}=-5-\sqrt{33}\end{align}$$
- step11: Solve the equation for $$x:$$
  $$\begin{align}&x=\log_{10}{\left(-5+\sqrt{33}ight)}\&x 
otin \mathbb{R}\end{align}$$
- step12: Find the union:
  $$x=\log_{10}{\left(-5+\sqrt{33}ight)}$$
The solution to the exponential equation $$100^{x}+10^{x+1}-8=0$$ is $$x=\log_{10}(-5+\sqrt{33})$$.

Therefore, the correct choice is A. The solution set is $$x=\log_{10}(-5+\sqrt{33})$$.
The solution is $$ x = \log_{10}(-5 + \sqrt{33}) $$.

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