- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,755

Find the values of $a$ and $b$.

- Thread starter anemone
- Start date

- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,755

Find the values of $a$ and $b$.

Find the values of $a$ and $b$.

is the answer a=0 b=-1

- Thread starter
- Admin
- #3

- Moderator
- #4

- Feb 7, 2012

- 2,725

So we have to look at the possibility $a = - \dfrac{b+1}{b+2}$. The equation $f(10a+6b+21) = 4\log2$ then says that $\log\left(-\tfrac{10(b+1)}{b+2} + 6b + 22\right) = \pm\log16$. Therefore $-\tfrac{10(b+1)}{b+2} + 6b + 22 = 16 $ or $\tfrac1{16}$.

If $-\tfrac{10(b+1)}{b+2} + 6b + 22 = 16 $ then $(b+1)\bigl(-10+ 6(b+2)\bigl) = 0$. That again has the unwanted solution $b=-1$, but it also has the solution $b+2 = \frac{10}6$, from which $b=-\frac13$. The corresponding value of $a$ is $-\frac25$.

(There was also the possibility that possibility that the equation $-\tfrac{10(b+1)}{b+2} + 6b + 22 = \tfrac1{16} $ might lead to a solution. But in fact that equation has no real solutions.) So the solution is $\boxed{a=-\dfrac25,\ b=-\dfrac13}$.