Solve the exponential equation. Express irrational solutions as decimals

A. The solect the correct choice below and, if necessary, fill in the answer box
(Round to the nearest thousandth as needed. Use a comma to
B. The solution is the empty set.

Question

Solve the exponential equation. Express irrational solutions as decimals

A. The solect the correct choice below and, if necessary, fill in the answer box
(Round to the nearest thousandth as needed. Use a comma to
B. The solution is the empty set.

UpStudy AI · Accepted Answer

Solve the equation $$ 5^{x-1}=3^{2x} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$5^{x-1}=3^{2x}$$
- step1: Take the logarithm of both sides:
  $$\log_{3}{\left(5^{x-1}ight)}=\log_{3}{\left(3^{2x}ight)}$$
- step2: Evaluate the logarithm:
  $$\left(x-1ight)	imes \log_{3}{\left(5ight)}=2x$$
- step3: Calculate:
  $$\log_{3}{\left(5ight)}	imes \left(x-1ight)=2x$$
- step4: Expand the expression:
  $$\log_{3}{\left(5ight)}	imes x-\log_{3}{\left(5ight)}=2x$$
- step5: Move the variable to the left side:
  $$\log_{3}{\left(5ight)}	imes x-\log_{3}{\left(5ight)}-2x=0$$
- step6: Collect like terms:
  $$\left(\log_{3}{\left(5ight)}-2ight)x-\log_{3}{\left(5ight)}=0$$
- step7: Move the constant to the right side:
  $$\left(\log_{3}{\left(5ight)}-2ight)x=0+\log_{3}{\left(5ight)}$$
- step8: Remove 0:
  $$\left(\log_{3}{\left(5ight)}-2ight)x=\log_{3}{\left(5ight)}$$
- step9: Divide both sides:
  $$\frac{\left(\log_{3}{\left(5ight)}-2ight)x}{\log_{3}{\left(5ight)}-2}=\frac{\log_{3}{\left(5ight)}}{\log_{3}{\left(5ight)}-2}$$
- step10: Divide the numbers:
  $$x=\log_{\frac{5}{9}}{\left(5ight)}$$
The solution to the exponential equation $$5^{x-1}=3^{2x}$$ is $$x \approx -2.738133$$.
The solution is $$x \approx -2.738$$.

Solve the exponential equation. Express irrational solutions as decimals

A. The solect the correct choice below and, if necessary, fill in the answer box
(Round to the nearest thousandth as needed. Use a comma to
B. The solution is the empty set.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions

Solve the exponential equation. Express irrational solutions as decimals A. The solect the correct choice below and, if necessary, fill in the answer box (Round to the nearest thousandth as needed. Use a comma to B. The solution is the empty set.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions

Solve the exponential equation. Express irrational solutions as decimals

A. The solect the correct choice below and, if necessary, fill in the answer box
(Round to the nearest thousandth as needed. Use a comma to
B. The solution is the empty set.