Solution assignment 08 Logarithmic equations

Solve:

$\ln[(x+8)(x-2)]=\ln(75)$

The domain of the function in the left-hand side is:

$x<-8$ or $x>2$

We rewrite the equation:

$(x+8)(x-2)=75$

$x^2+6x-16=75$

$x^2+6x-91=0$

The solution of this equation is:

$x_{1,2}=\displaystyle\frac{-6\pm20}{2}$

thus:

$x_1=-13$ or $x_2=7$

Both solutions lie in the domain and are valid solutions.

Note. Compare this equation with the following assignment.

Solve:

$\ln(x+8)+\ln(x-2)=\ln(75)$

In this case the domain of the function in the left-hand side is:

$x>-8$ and $x>2$

thus:

$x>2$

We proceed in the same way as before, with as result the solutions:

$x_1=-13$ or $x_2=7$

However, now the first solution does not satisfy the equation because it lies outside the domain.