Question

Solve the equation $3^{5x}·{81}^{1-x}=9^{x-3}$. Check your solution.

Short answer

The solution for the equation is $x=10$.

Step-by-step solution

Step 1. Write down the given equation.

The given equation is $3^{5x}·{81}^{1-x}=9^{x-3}$.

Step 2. Simplify the equation.

The given equation $3^{5x}·{81}^{1-x}=9^{x-3}$is solved as:

$\begin{array}{l}{3}^{5x}\cdot {81}^{1-x}=9^{x-3}\mathrm{....}\left(\text{Given}\right)\\ {\left(3\right)}^{5x}\cdot {\left(3^4\right)}^{1-x}={\left(3^2\right)}^{x-3}\\ {\left(3\right)}^{5x}\cdot {\left(3\right)}^{4-4x}={\left(3\right)}^{2x-6}\\ {\left(3\right)}^{5x+4-4x}={\left(3\right)}^{2x-6}\\ {\left(3\right)}^{x+4}={\left(3\right)}^{2x-6}\\ x+4=2x-6\\ x=10\end{array}$

Step 3. Verify the result for obtained value.

Plugging $x=10$in given equation and verify the result.

$\begin{array}{c}{3}^{5x}\cdot {81}^{1-x}=9^{x-3}\mathrm{....}\left(\text{Given}\right)\\ 3^{5\left(10\right)}\cdot {81}^{1-\left(10\right)}=9^{\left(10\right)-3}\\ 3^{50}\cdot {81}^{-9}=9^7\\ {\left(3\right)}^{50}\cdot {\left(3^4\right)}^{-9}={\left(3^2\right)}^7\\ {\left(3\right)}^{50}\cdot {\left(3\right)}^{-36}={\left(3\right)}^{14}\\ {\left(3\right)}^{14}={\left(3\right)}^{14}\mathrm{....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $x=10$.

Step 4. Conclusion.

The solution for the given equation is $x=10$.