Lesson 6.4 : Graphs of Logarithmic Functions

New Concept

Explore the graphs of logarithmic functions, the inverses of exponential functions. You'll learn to identify their domain and graph the parent function $y = \operatorname{log}_b(x)$, along with its various transformations like shifts, stretches, and reflections.

What’s next

You're ready to start! Next, you will tackle a series of practice cards and interactive examples to master graphing logarithmic functions and their transformations.

Property

The domain of a logarithmic function consists only of positive real numbers. The argument of the logarithmic function must be greater than zero. To find the domain:

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

Examples

• To find the domain of $f(x) = \operatorname{log}_3(x - 5)$, set the argument $x - 5 > 0$. Solving for $x$ gives $x > 5$. The domain is $(5, \infty)$.
• For $g(x) = \operatorname{log}_7(4 - 2x)$, the argument must be positive: $4 - 2x > 0$. This simplifies to $-2x > -4$, or $x < 2$. The domain is $(-\infty, 2)$.
• The domain of $h(x) = \operatorname{ln}(3x + 9)$ requires $3x + 9 > 0$. Solving this gives $3x > -9$, so $x > -3$. The domain is $(-3, \infty)$.

Explanation

Logarithms can't process zero or negative inputs. Think of the argument inside the parentheses as needing to be strictly positive. Solving the inequality argument > 0 reveals the function's valid x values, which is its domain.

Property

For any real number $x$ and constant $b > 0, b \neq 1$, the graph of $f(x) = \operatorname{log}_b(x)$ has these characteristics:

• one-to-one function
• vertical asymptote: $x = 0$
• domain: $(0, \infty)$
• range: $(-\infty, \infty)$
• $x$-intercept: $(1, 0)$ and key point $(b, 1)$
• $y$-intercept: none
• increasing if $b > 1$
• decreasing if $0 < b < 1$

Examples

• The graph of $f(x) = \operatorname{log}_4(x)$ is increasing because the base is $b=4 > 1$. It has an x-intercept at $(1, 0)$, a key point at $(4, 1)$, and a vertical asymptote at $x=0$.
• The graph of $g(x) = \operatorname{log}_{1/3}(x)$ is decreasing since the base is $b=1/3 < 1$. It has an x-intercept at $(1, 0)$, a key point at $(\frac{1}{3}, 1)$, and a vertical asymptote at $x=0$.
• The graph of $h(x) = \operatorname{ln}(x)$ has base $e \approx 2.718$, so it is increasing. It passes through the x-intercept $(1, 0)$ and the key point $(e, 1)$, with a vertical asymptote at $x=0$.

Explanation

Every parent log graph has a vertical 'wall' (asymptote) at $x=0$ and crosses the x-axis at $(1, 0)$. If the base $b$ is greater than 1, the graph climbs up; if $b$ is between 0 and 1, it slides down.

Property

For any constant $c$, the function $f(x) = \operatorname{log}_b(x + c)$:

• shifts the parent function $y = \operatorname{log}_b(x)$ left $c$ units if $c > 0$.
• shifts the parent function $y = \operatorname{log}_b(x)$ right $c$ units if $c < 0$.
• has the vertical asymptote $x = -c$.

For any constant $d$, the function $f(x) = \operatorname{log}_b(x) + d$:

• shifts the parent function $y = \operatorname{log}_b(x)$ up $d$ units if $d > 0$.
• shifts the parent function $y = \operatorname{log}_b(x)$ down $d$ units if $d < 0$.

Examples

• The function $f(x) = \operatorname{log}_2(x - 3)$ shifts the parent graph 3 units to the right. Its new vertical asymptote is $x = 3$.
• The function $g(x) = \operatorname{log}_5(x) + 4$ shifts the parent graph 4 units up. Its vertical asymptote remains at $x = 0$.
• The function $h(x) = \operatorname{log}_7(x + 1) - 2$ is shifted 1 unit to the left and 2 units down. Its vertical asymptote is $x = -1$.

Property

The function $f(x) = a\operatorname{log}_b(x)$:

• stretches the parent function $y = \operatorname{log}_b(x)$ vertically by a factor of $a$ if $|a| > 1$.
• compresses the parent function $y = \operatorname{log}_b(x)$ vertically by a factor of $a$ if $0 < |a| < 1$.
• reflects the parent function about the $x$-axis if $a < 0$.

The function $f(x) = \operatorname{log}_b(-x)$:

• reflects the parent function $y = \operatorname{log}_b(x)$ about the $y$-axis.
• has domain $(-\infty, 0)$.

Examples

• The graph of $f(x) = 3\operatorname{log}_4(x)$ is a vertical stretch of its parent graph by a factor of 3. The key point $(4, 1)$ is transformed to $(4, 3)$.
• The graph of $g(x) = -\operatorname{log}_2(x)$ is a reflection of its parent graph across the x-axis. The key point $(2, 1)$ is transformed to $(2, -1)$.
• The graph of $h(x) = \operatorname{log}_5(-x)$ is a reflection of its parent graph across the y-axis. Its domain changes to $(-\infty, 0)$ and its x-intercept becomes $(-1, 0)$.