Solve the equation. Round to the nearest hundredth. Show work. 8^(-x+7) = 3^(7x+2)

Question

Solve the equation. Round to the nearest hundredth. Show work.


`8^(-x+7) = 3^(7x+2)`

Answer 1

Final answer is approx x=1.27.

Explanation:

Given equation is
`8^(-x+7)=3^(7x+2)`.

Now we need to solve equation
`8^(-x+7)=3^(7x+2)` and round to the nearest hundredth.

`8^(-x+7)=3^(7x+2)`

`\log(8^(-x+7))=\log(3^(7x+2))`

`\left(-x+7\right)\cdot\log\left(8\right)=\left(7x+2\right)\cdot\log\left(3\right)`

`-x\cdot\log\left(8\right)+7\cdot\log\left(8\right)=7x\cdot\log\left(3\right)+2\cdot\log\left(3\right)`

`-x\cdot\log\left(8\right)-7x\cdot\log\left(3\right)=2\cdot\log\left(3\right)-7\cdot\log\left(8\right)`

`x\left(-\log\left(8\right)-7\cdot\log\left(3\right)\right)=\left(2\cdot\log\left(3\right)-7\cdot\log\left(8\right)\right)`

`x=(\left(2\cdot\log\left(3\right)-7\cdot\log\left(8\right)\right))/(\left(-\log\left(8\right)-7\cdot\log\left(3\right)\right))`

Now use calculator to calculate log values, we get:

`x=1.26501646392`

Round to the nearest hundredth.

Hence final answer is approx x=1.27.