How do I find the domain of the logarithmic function? See photo

Question

asked Oct 7, 2023 188k views

1 Answer

Domness · Answer 1 · 2023-10-11T16:46:23+0000

Answer:

$(-\infty,-5)\cup(4,\infty)$

Step-by-step explanation:

Given the below function;

$\log ((x+5)/(x-4))$

Recall that the domain of a function is a set of input values for which the function is defined.

Note that a logarithmic function is only defined when the input is positive, so the given function is defined when x + 5 > 0 and x - 4 > 0.

Let's go ahead and solve the inequalities as seen below;

$\begin{gathered} x+5>0 \\ x<-5 \\ \text{And } \\ x-4>0 \\ x>4 \end{gathered}$

Notice that for the function to be defined also, the denominator must not be equal to zero, so we'll have;

$\begin{gathered} x-4\\e0 \\ x\\e4 \end{gathered}$

We can see that for the given function to be defined, the following restrictions have to be considered;

$\begin{gathered} x<-5 \\ x>4 \\ x\\e4 \end{gathered}$

Therefore the domain of the given function can be written in interval notation as;

$(-\infty,-5)\cup(4,\infty)$