AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsLogarithmic FunctionsKey Characteristics of Logarithmic Functions

Key Characteristics of Logarithmic Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Key characteristics of logarithmic functions

What are the domain and range of a logarithmic function?

For a logarithmic function in general form $f (x) = a \log_{b} x$
- The domain is all positive real numbers, $x > 0$
- The range is all real numbers
The domain restriction exists because you can only take the logarithm of a positive number
- $\log_{b} x$ is not defined when $x \leq 0$
Compare this to the general exponential function $a b^{x}$ , which has the opposite features
- its domain is all real numbers
- its range is only positive values (when $a > 0$ )
This swap of domain and range reflects the inverse relationship between the two function types

Is a logarithmic function always increasing or always decreasing?

Because logarithmic functions are inverses of exponential functions
- they share a similar "one-direction" behavior
A logarithmic function $f (x) = a \log_{b} x$ is always increasing or always decreasing
- It never changes direction
When $a > 0$ and $b > 1$
- the function is always increasing
When $a > 0$ and $0 < b < 1$
- the function is always decreasing
When $a < 0$ , these are reversed
- i.e. decreasing when $b > 1$
- and increasing when $0 < b < 1$

The graph is also always concave up or always concave down
- The concavity never changes
As a consequence:
- Logarithmic functions have no extrema (no maximum or minimum values)
  - unless the domain is restricted to a closed interval
- Their graphs have no points of inflection
These are the same properties that exponential functions have
- Both function types exhibit this consistent, one-directional behavior because of their proportional structure

What are the end behavior and asymptotic behavior of a logarithmic function?

Logarithmic functions in general form have a vertical asymptote at $x = 0$
- As $x$ approaches 0 from the right
  - the function values decrease or increase without bound (depending on the signs of $a$ and $b$ )
As $x$ increases without bound, the function values also increase or decrease without bound
- but they do so very slowly
In limit notation (for the most common case with $a > 0$ , $b > 1$ )
- $\lim_{x \to 0^{+}} a \log_{b} x = - \infty$
- $\lim_{x \to \infty} a \log_{b} x = \infty$

Graph of y = log₂(x) showing a curve passing through point (1,0), increasing to the right with a vertical asymptote at x=0, and labels for limits. — **Graph of a logarithmic function**

Compare this to exponential functions, which have a horizontal asymptote
- The swap from horizontal to vertical asymptote is another consequence of the inverse relationship
- Reflecting over the line $y = x$ turns horizontal asymptotes into vertical ones

Examiner Tips and Tricks

Limit notation for logarithmic end behavior is assessed on the exam.

Make sure you are comfortable writing expressions like $\lim_{x \to 0^{+}} f (x) = - \infty$ and $\lim_{x \to \infty} f (x) = \infty$ .

Note the " $0^{+}$ " notation in the first limit, indicating approach from the right (positive side) only
This is necessary since the function is not defined for $x \leq 0$

How can you identify a 'hidden' logarithmic function?

Just as with exponential functions, data may not immediately appear to be logarithmic because of a shift
For a logarithmic function in general form $f (x) = a \log_{b} x$
- the input values change proportionally
  - over equal-length output-value intervals
- This is the "reversed" version of the exponential property, where output values change proportionally over equal-length input intervals
If an additive transformation $g (x) = f (x + k)$ (where $k \neq 0$ ) is applied to a logarithmic function $f$ in general form
- the transformed function $g$ no longer has this proportional-input property
However, if you encounter any function where an additive transformation does produce proportional input values over equal-length output intervals
- that's a signal that the function can be modeled as a translation of a logarithmic function
E.g. consider the following table of input-output values

$x$	5	8	14	26	50
$g (x)$	3	4	5	6	7

The outputs increase by 1 each time
- and the corresponding inputs are $5, 8, 14, 26, 50$
The inputs are not proportional
- $\frac{8}{5} \neq \frac{14}{8} \neq \frac{26}{14} \neq \frac{50}{26}$
But subtracting 2 from each input gives $3, 6, 12, 24, 48$
- and $\frac{6}{3} = \frac{12}{6} = \frac{24}{12} = \frac{48}{24} = 2$ , which is a constant ratio
So the data in the table can be modeled by a horizontal translation of a logarithmic function
- I.e. $g (x) = f (x - 2)$
  - where $f (x)$ is a logarithmic function
  - possibly with a vertical translation as well to produce those precise output values

Examiner Tips and Tricks

The characteristics of logarithmic functions mirror those of exponential functions in many ways.

If you understand one, you can often deduce the other by thinking about the inverse relationship.

swapping inputs and outputs
swapping domain and range
swapping horizontal and vertical asymptotes

Worked Example

The function $f$ is given by $f (x) = - 3 \log_{5} x$ .

(a) State the domain of $f$ .

Answer:

The domain of $f$ is $x > 0$ (all positive real numbers), since the input to a logarithm must be positive

(b) Determine whether $f$ is increasing or decreasing. Justify your answer.

Answer:

The base is $b = 5 > 1$ , so $\log_{5} x$ is an increasing function

However, the coefficient $a = - 3$ is negative, which reflects the graph over the $x$ -axis and reverses the direction

Therefore $f$ is decreasing

(c) Describe the end behavior of $f$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.

Answer:

As $x$ increases without bound, $\log_{5} x$ increases without bound

so $- 3 \log_{5} x$ decreases without bound.

$\lim_{x \to \infty} f (x) = - \infty$

(d) Describe the behavior of $f$ near its vertical asymptote. Express your answer using the mathematical notation of a limit.

Answer:

The function $f$ has a vertical asymptote at $x = 0$

As $x$ approaches 0 from the right, $\log_{5} x$ decreases without bound (becomes very negative)

so $- 3 \log_{5} x$ increases without bound (becomes very positive)

$\lim_{x \to 0^{+}} f (x) = \infty$

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Key Characteristics of Logarithmic Functions (College Board AP® Precalculus): Revision Note

Key characteristics of logarithmic functions

What are the domain and range of a logarithmic function?

Is a logarithmic function always increasing or always decreasing?

What are the end behavior and asymptotic behavior of a logarithmic function?

Examiner Tips and Tricks

How can you identify a 'hidden' logarithmic function?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis