Question

Use the equation, (1/27)^x=3^(-4x+6), to complete the following problems.

Rewrite the equation using the same base.
Solve for x. Write your answer as a fraction in simplest form.
Please show all work, and refrain from posting links, thank you!

Answer 1

Given equation:

`\left((1)/(27)\right)^x=3^((-4x+6))`

27 can be written as
`3^3`

Also
`(1)/(a^b)` can be written as
`a^(-b)`

`\implies (1)/(27)=(1)/(3^3)=3^(-3)`

Therefore, we can rewrite the given equation with base 3:

`\implies (3^(-3))^x=3^((-4x+6))`

To solve, apply the exponent rule
`(a^b)^c=a^(bc)`

`\implies 3^(-3 \cdot x)=3^((-4x+6))`

`\implies 3^((-3x))=3^((-4x+6))`

`\textsf{If }a^(f(x))=a^(g(x)), \textsf{ then } f(x)=g(x)`

`\implies -3x=-4x+6`

Add
`4x` to both sides:

`\implies x=6`