Question

Can someone solve this equation please? `3^(x+1)=243^(x+2)`

Answer 1

`x =-9/4`

Explanation:

This is an exponential equation where the variable is in the index.

The first step s to make the bases the same.

It helps to know that `3^5 = 243`

`3^(x+1) = color(blue)(243)^(x+2)`
`color(white)(wwwww)darr`
`3^(x+1) = (color(blue)(3^5))^(x+2)`

`3^(x+1) = 3^(5x+10)`

If the bases are equal then the indices must be equal.

`:. x+1 = 5x+10`

`1-10 = 5x-x`

`-9 = 4x`

`-9/4 = x`

Answer 2

`x=-2.25 or -9/4`

Explanation:

As an addition to EZ as pi's response, you can use logarithms as a more general approach. Logarithms will work with most exponential equations.

Take the log of both sides:

`log3^(x+1)=log243^(x+2) =>`

`(x+1)log3=(x+2)log243=>`

`(x+1)0.47712=(x+2)2.3856=>`

`0.47712x+0.47712=2.3856x+4.7712`

`-1.90848x=4.29408`

`x=-2.25 or -9/4`