Question

9^x= e^x+8
Solve for x using logs

Answer 1

`x=(8)/(\ln(9)-1)`

Explanation:

I will make an assumption.

We are trying to solve:

`9^x=e^(x+8)`

We first need to have the same base on both sides. So I'm going to write
`9^x` as
`e^(\ln(9^x))` using that natural log (ln( )) and e^() are inverses.

So we are going to rewrite the equation:

`e^(\ln(9^x))=e^(x+8)`

So now that the bases are the same we can set the exponents equal like so:

`\ln(9^x)=x+8`

Using power rule we can bring down the x in front of ln( ):

`x\ln(9)=x+8`

Now we need to get our terms with x on one side and terms without x on the opposite side. To do this, we just need to subtract x on both sides in this case:

`x\ln(9)-x=8`

Factor the left hand side like:

`x(\ln(9)-1)=8`

Now divide both sides by what x is being multiplied by:

`x=(8)/(\ln(9)-1)`