Identify the domain of a logarithmic function

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined as $y={b}^{x}$ for any real number x and constant $b>0$, $b\ne 1$, where

• The domain of y is $\left(-\infty ,\infty \right)$.
• The range of y is $\left(0,\infty \right)$.

In the last section we learned that the logarithmic function $y={\mathrm{log}}_{b}\left(x\right)$ is the inverse of the exponential function $y={b}^{x}$. So, as inverse functions:

• The domain of $y={\mathrm{log}}_{b}\left(x\right)$ is the range of $y={b}^{x}$:$\left(0,\infty \right)$.
• The range of $y={\mathrm{log}}_{b}\left(x\right)$ is the domain of $y={b}^{x}$: $\left(-\infty ,\infty \right)$.

Transformations of the parent function $y={\mathrm{log}}_{b}\left(x\right)$ behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape.

In Graphs of Exponential Functions we saw that certain transformations can change the range of $y={b}^{x}$. Similarly, applying transformations to the parent function $y={\mathrm{log}}_{b}\left(x\right)$ can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.

For example, consider $f\left(x\right)={\mathrm{log}}_{4}\left(2x - 3\right)$. This function is defined for any values of x such that the argument, in this case $2x - 3$, is greater than zero. To find the domain, we set up an inequality and solve for x:

$\begin{cases}2x - 3>0\hfill & \text{Show the argument greater than zero}.\hfill \\ 2x>3\hfill & \text{Add 3}.\hfill \\ x>1.5\hfill & \text{Divide by 2}.\hfill \end{cases}$

In interval notation, the domain of $f\left(x\right)={\mathrm{log}}_{4}\left(2x - 3\right)$ is $\left(1.5,\infty \right)$.

How To: Given a logarithmic function, identify the domain.

1. Set up an inequality showing the argument greater than zero.
2. Solve for x.
3. Write the domain in interval notation.

Example 1: Identifying the Domain of a Logarithmic Shift

What is the domain of $f\left(x\right)={\mathrm{log}}_{2}\left(x+3\right)$?

Solution

The logarithmic function is defined only when the input is positive, so this function is defined when $x+3>0$. Solving this inequality,

$\begin{cases}x+3>0\hfill & \text{The input must be positive}.\hfill \\ x>-3\hfill & \text{Subtract 3}.\hfill \end{cases}$

The domain of $f\left(x\right)={\mathrm{log}}_{2}\left(x+3\right)$ is $\left(-3,\infty \right)$.

Try It 1

What is the domain of $f\left(x\right)={\mathrm{log}}_{5}\left(x - 2\right)+1$?

Solution

Example 2: Identifying the Domain of a Logarithmic Shift and Reflection

What is the domain of $f\left(x\right)=\mathrm{log}\left(5 - 2x\right)$?

Solution

The logarithmic function is defined only when the input is positive, so this function is defined when $5 - 2x>0$. Solving this inequality,

$\begin{cases}5 - 2x>0\hfill & \text{The input must be positive}.\hfill \\ -2x>-5\hfill & \text{Subtract }5.\hfill \\ x<\frac{5}{2}\hfill & \text{Divide by }-2\text{ and switch the inequality}.\hfill \end{cases}$

The domain of $f\left(x\right)=\mathrm{log}\left(5 - 2x\right)$ is $\left(-\infty ,\frac{5}{2}\right)$.

Try It 2

What is the domain of $f\left(x\right)=\mathrm{log}\left(x - 5\right)+2$?

Solution