Question

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

Answer 1

`(-\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)`