# Range and Domain of Logarithmic Function

One of the easiest topics to comprehend is logarithm specifically the range and domain of logarithmic function. The graph of a logarithmic function is the reflection of an exponential function. The range and domain of the graph of this function is critical to understand how this function behaves. Given a logarithmic function $f(x)=\log_b U$ where b>1 and U>0. Any values of $b\leq 1$ and $U\leq 0$ the function does not exist.

Sample Problem 1:

Find the range and domain of $y=\log (x-2)$

Solution

The range of the function is the set of all real numbers, in symbol $(-\infty,+\infty)$

For the domain of the function we just need to focus on U in logU. In this example, $U=x-2$

Since U>0, or x-2>0 or x>2. In symbols $(2,+\infty)$

Sample Problem 2:

Find the range and domain of $y=\log (x+1)$

Solution

Range: $(-\infty,+\infty)$

Domain: $(-1, +\infty)$ the explanation is the same as mentioned above.

Sample Problem 3:

Find the range and domain of $y=\log (9-x^2)$

Solution

Range: $(-\infty,+\infty)$

Domain: $9-x^2$>0. or $x^2-9$<0. This time, you need to have a skill to solve quadratic inequality. The roots are -3 and 3 and in this form of quadratic equation the solution set or the Domain is (-3,3)

Sample Problem 4:

Find the range and domain of $y=\log (x^2-3x+2)$.

Solution

Range: $(-\infty,+\infty)$

Domain: $x^2-3x+2$>0. The roots of this equation are 1 and 2. The solution set of this quadratic inequality is $(-\infty,1)\cup (2,+\infty)$. This is also the domain.

Challenge:

Direction: Find the Range and Domain of the following

1. $y=\log(x+9)$

2. $y=\log (x-5)$

3. $y=\log (x^2-36)$

4. $y=\log (1-x^2)$

5. $y=\log (x^2-2x+1)$

1. Range: $(-\infty,+\infty)$;    Domain: $(-9,+\infty)$

2. Range: $(-\infty,+\infty)$;    Domain: $(5,+\infty)$

3. Range: $(-\infty,+\infty)$;    Domain: $(-\infty,-6)\cup (6,+\infty)$

4. Range: $(-\infty,+\infty)$;     Domain: $(-1,1)$

5. Range: $(-\infty,+\infty)$;     Domain: $x\ne 1$