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## Q. 3.2.10

Finding Domains of Natural Logarithmic Functions

Find the domain of each function:

a. f(x) = ln(3 x)                b. h(x) = ln(x 3)².

## Verified Solution

a. The domain of f consists of all x for which 3 – x > 0. Solving this inequality for x, we obtain x < 3. Thus, the domain of f is $\{x \mid x<3\}$ or (-∞, 3). This is verified by the graph in Figure 3.14.

b. The domain of h consists of all x for which $(x-3)^2>0$. It follows that the domain of h is all real numbers except 3 . Thus, the domain of h is $\{x \mid x \neq 3\}$ or $(-∞, 3) \cup(3, \infty)$. This is shown by the graph in Figure 3.15. To make it more obvious that 3 is excluded from the domain, we used a DOT format.