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.

logx

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)

 

Answer key:

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

 

 

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