Question

How do you solve `log_12 (p^2-5p)=log_12 (8+2p)`?

Answer 1

Hi there! To solve this, you must recognize that if you have logs of the same base, you can drop them, leaving you with the functions inside.

Explanation:

When you have:

`log_b(f(x)) = log_b(g(x))` this is equivalent to:

`f(x) = g(x)`

Since both logs in your question have a base of 12, you can drop them, leaving you with:

`p^2 -5p = 8 + 2p`

Rearranging we get:

`p^2 - 5p -2p - 8 = 0`

Collecting like terms:

`p^2 - 7p - 8 = 0`

Factoring this simple trinomial (finding numbers that multiply to -8 and add to -7):

`(p-8)(p+1)=0`

Solving each piece we get:

`p=8,-1`

And that's it, those are the p values that would make those expressions equal. Hopefully everything was clear and helpful! If you have any questions, please feel free to ask! :)

Answer 2

`p=-1 and p = 8`.

Explanation:

Both the logarithms have the same base 12.
So, `p^2-5p=8+2p`.
`p^2-7p-8=0`
The roots of this quadratic equation are`-1 and 8.`.
Both the roots are admissible for both the logarithms..