Teacher Guide: Graphing Logarithmic Functions

Learn to graph logarithmic functions: key points, domain, range, the vertical asymptote, the mirror image of the exponential function and shifts.

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10 questions on Logarithms. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Graph $y = \log_b x$ for $b > 1$ and $0 < b < 1$ using the key points
State the domain, range, vertical asymptote and intercepts of a logarithmic function
Explain the graph of $y = \log_b x$ as the reflection of $y = b^x$ across $y = x$
Graph shifts and reflections of $y = \log_b x$ , such as $y = -\log_3(x + 2)$
Find the equation $y = \log_b(x - h) + k$ from an asymptote and two points

Prerequisites

• Definition of a Logarithm: $\log_b y = x \iff b^x = y$
• Graphs of exponential functions $y = b^x$
• Shifting and reflecting graphs of functions
• Common Logarithms (Base 10) and Natural Logarithms (ln)

Discussion Starters

1. Why can the graph of $y = \log_2 x$ never reach the $y$ -axis, even though it gets very close?
2. The curve looks almost flat for large x. Does it ever stop rising? How could you convince a friend?
3. What happens to the points $(1, 0)$ and $(b, 1)$ when you change $h$ and $k$ ?
4. Why does the reflection across $y = x$ turn a horizontal asymptote into a vertical one?

Common Misconceptions

$y = \log_b(x - h)$ moves the graph to the left when $h$ is positive

Remediation: Have students solve $x - h = 1$ to find where the graph crosses $y = 0$ : for $h = 3$ it is $x = 4$ , three units to the right of $(1, 0)$ .

The log graph has a horizontal asymptote because it looks flat

Remediation: Make a table with $x = 2, 4, 8, \dots, 1024$ for $y = \log_2 x$ : the y-values $1, 2, 3, \dots, 10$ keep growing by $1$ each time.

A base between 0 and 1 gives a graph that is not a logarithm or is undefined

Remediation: Compute $\log_{1/2} 4 = -2$ and $\log_{1/2} \frac{1}{2} = 1$ with the definition and plot them: the curve is a decreasing mirror image of $y = \log_2 x$ .

Differentiation Ideas

For Struggling Students:

• Start with $y = \log_2 x$ and a table of powers of 2 only
• Use the three key points $\left(\frac{1}{b}, -1\right)$ , $(1, 0)$ , $(b, 1)$ as a fixed routine for every graph
• Shift one feature at a time: first $k$ , then $h$

For On-Level Students:

• Graph functions such as $y = \log_3(x + 2) - 1$ and state the domain and asymptote
• Compare $y = \log_2 x$ and $y = \log_{1/2} x$ in the grapher
• Find the equation of a log graph from its asymptote and two points

For Advanced Students:

• Graph $y = \log_2(6 - 2x)$ and explain why its domain is $x < 3$
• Investigate $y = a \log_b x$ for different $a$ and compare with changing the base
• Prove that $y = \log_{1/b} x$ is the reflection of $y = \log_b x$ across the $x$ -axis

Standards Alignment

HSF.IF.C.7e (CCSS.MATH.CONTENT.HSF.IF.C.7.E)
Graph exponential and logarithmic functions, showing intercepts and end behavior
HSF.BF.B.3 (CCSS.MATH.CONTENT.HSF.BF.B.3)
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx) and f(x + k)
HSF.BF.B.5 (CCSS.MATH.CONTENT.HSF.BF.B.5)
Understand the inverse relationship between exponents and logarithms

Lesson Resources

visualLog Graph Explorer
Sliders for $b$ , $h$ and $k$ move the curve, the asymptote and the two key points live
activityFold Along y = x
Students graph $y = 2^x$ on tracing paper, fold along $y = x$ and trace the logarithmic graph
worksheetAsymptote, Domain, Points
Ten transformed log functions: find the asymptote, domain and two key points, then sketch

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A logarithmic function has the form

f(x) = \log_b x

with base

b > 0

and

b \neq 1

. It is the inverse function of the exponential function

y = b^x

y = \log_b x

means

b^y = x

Key points (they come straight from the definition):

$\left(\frac{1}{b}, -1\right)$ because $b^{-1} = \frac{1}{b}$
$(1, 0)$ because $b^0 = 1$
$(b, 1)$ because $b^1 = b$

**Features of the graph of

y = \log_b x

:**

Domain: $x > 0$ (you can only take the logarithm of a positive number)
Range: all real numbers
Vertical asymptote: the line $x = 0$ (the $y$ -axis); the graph gets closer and closer to it but never touches it
$x$ -intercept**: $(1, 0)$ ; there is no $y$ -intercept
If $b > 1$ , the graph increases; if $0 < b < 1$ , it decreases

The graph of

y = \log_b x

is the reflection of the graph of

y = b^x

across the line

y = x

: every point

(p, q)

y = b^x

becomes the point

(q, p)

Transformations:

y = \log_b(x - h) + k

shifts the graph $h$ units right (left if $h < 0$ ) and $k$ units up (down if $k < 0$ )
vertical asymptote $x = h$ , domain $x > h$ , key points $(1 + h, k)$ and $(b + h, 1 + k)$
$y = -\log_b x$ reflects the graph across the $x$ -axis; $y = \log_b(-x)$ reflects it across the $y$ -axis (domain $x < 0$ )

In the interactive graph, move the sliders

b

h

and

k

and watch the asymptote and the two key points move.

Worked Examples

Graph $y = \log_2 x$ and state its domain, range and asymptote.

Choose x-values that are powers of 2

$x = \frac{1}{4}, \frac{1}{2}, 1, 2, 4, 8$ , because $\log_2$ of a power of 2 is easy to find → Six easy x-values

Find each y-value

$\log_2 \frac{1}{4} = -2$ , $\log_2 \frac{1}{2} = -1$ , $\log_2 1 = 0$ , $\log_2 2 = 1$ , $\log_2 4 = 2$ , $\log_2 8 = 3$ → $y = -2, -1, 0, 1, 2, 3$

Plot the points and draw the asymptote

Plot $\left(\frac{1}{4}, -2\right)$ , $\left(\frac{1}{2}, -1\right)$ , $(1, 0)$ , $(2, 1)$ , $(4, 2)$ , $(8, 3)$ and draw the dashed line $x = 0$ → Points and asymptote $x = 0$

Connect with a smooth curve

The curve falls steeply near the $y$ -axis without touching it and rises slowly to the right → Increasing curve through $(1, 0)$

Answer: Domain $x > 0$ , range all real numbers, vertical asymptote $x = 0$ , $x$ -intercept $(1, 0)$ .

Common Mistakes

Shifting $y = \log_2(x - 3)$ three units to the left

Why it's wrong: The minus sign looks like a move to the left, but the argument $x - 3$ must be positive, which means $x > 3$ : the whole graph lies to the right of $x = 3$ .

Correct: $y = \log_b(x - h)$ moves the graph $h$ units to the right. $y = \log_2(x - 3)$ has the asymptote $x = 3$ .

Letting the graph touch or cross the $y$ -axis

Why it's wrong: $\log_b 0$ and $\log_b$ of a negative number do not exist, because $b^y$ is always positive.

Correct: The graph of $y = \log_b x$ comes closer and closer to $x = 0$ but never reaches it. The domain is $x > 0$ .

Thinking the graph levels off at some height

Why it's wrong: The curve rises slowly, so it looks flat, but it never stops rising: $\log_2 1024 = 10$ and $\log_2 1048576 = 20$ .

Correct: A logarithmic graph has no horizontal asymptote. The range is all real numbers.

Keeping the domain $x > 0$ after a horizontal shift

Why it's wrong: The domain comes from the argument, not from $x$ itself.

Correct: For $y = \log_3(x + 5)$ the argument $x + 5$ must be positive, so the domain is $x > -5$ .

Why It Matters

Graphs of logarithmic functions show how a quantity grows when every new step needs many times more input:

Sound: a sound level meter shows $L = 10 \log\left(\frac{I}{I_0}\right)$ decibels; multiplying the intensity by $10$ adds only $10$ dB, which is why the curve is so flat.
Chemistry: $\text{pH} = -\log[H^+]$ is a reflected log graph: more hydrogen ions give a smaller pH.
Computer science: binary search needs about $\log_2 n$ steps for $n$ items; doubling the list adds only one step.
Earthquakes: the Richter scale grows by $1$ every time the amplitude is multiplied by $10$ .

Reading these graphs means knowing three things at a glance: where the asymptote is, which values are allowed (domain), and how fast the curve rises. This lesson builds on Definition of a Logarithm and prepares you for Solving Logarithmic Equations.

Real World Applications

Sound Level in Decibels

A sound level meter shows $L = 10 \log\left(\frac{I}{I_0}\right)$ decibels, where $\frac{I}{I_0}$ tells how many times stronger the sound is than the quietest sound we can hear. The graph is a stretched log curve: very flat for large intensities.

Example:

A sound $100$ times stronger than $I_0$ gives $L = 10 \log 100 = 20$ dB. A sound $10$ times stronger again gives only $30$ dB.

1Try It Yourself

A quiet library is $1000$ times stronger than the quietest audible sound, so $\frac{I}{I_0} = 1000$ .

What is the sound level $L$ in decibels?

Step 1: Write the mathematical expression

Calculate $L = 10 \log(1000)$ :

Binary Search in Computer Science

Binary search finds a name in a sorted list by halving the list again and again. For $n$ names it needs about $\log_2 n$ halvings, so the graph of the number of steps is the graph of $y = \log_2 x$.

Example:

A list of $8$ names needs $\log_2 8 = 3$ halvings: $8 \to 4 \to 2 \to 1$ .

2Try It Yourself

A sorted contact list has $1024$ names. Each step of binary search halves the list.

How many halvings are needed to get down to one name?

Step 1: Write the mathematical expression

Calculate $\log_2(1024)$ :

The pH Scale in Chemistry

Acidity is measured with $\text{pH} = -\log[H^+]$, where $[H^+]$ is the hydrogen ion concentration. The minus sign reflects the log graph across the $x$-axis, so the pH goes down when the concentration goes up.

Example:

Lemon juice has $[H^+] = 0.01 = 10^{-2}$ , so $\text{pH} = -\log(10^{-2}) = 2$ .

3Try It Yourself

A sample of rainwater has $[H^+] = 0.0001$ .

What is its pH?

Step 1: Write the mathematical expression

Calculate $\text{pH} = -\log(0.0001)$ :

Key Takeaways

1 $y = \log_b x$ is the inverse of $y = b^x$ : its graph is the reflection of $y = b^x$ across the line $y = x$
2Key points of $y = \log_b x$ : $\left(\frac{1}{b}, -1\right)$ , $(1, 0)$ and $(b, 1)$
3Domain $x > 0$ , range all real numbers, vertical asymptote $x = 0$ , no $y$ -intercept
4For $b > 1$ the graph increases; for $0 < b < 1$ it decreases
5 $y = \log_b(x - h) + k$ moves the graph $h$ right and $k$ up: asymptote $x = h$ , domain $x > h$
6A minus sign in front of the log reflects the graph across the $x$ -axis; $\log_b(-x)$ reflects it across the $y$ -axis

Frequently Asked Questions

Why does the graph never touch the $y$ -axis?

On the

y

-axis

x = 0

, and

\log_b 0

would need

b^y = 0

. A positive base raised to any power is positive, so no

y

works. Near

x = 0

the values get very large negative numbers (for

b > 1

), which is why the curve dives along the asymptote.

Does the log graph have a horizontal asymptote like the exponential graph?

No. The exponential graph has the horizontal asymptote

y = 0

; after the reflection across

y = x

it becomes the vertical asymptote

x = 0

. The log graph keeps rising (for

b > 1

), only very slowly.

What do the graphs of $\log x$ and $\ln x$ look like?

Both have a base greater than 1 (

10

and

e \approx 2.718

), so both increase, pass through

(1, 0)

and have the asymptote

x = 0

\ln x

passes through

(e, 1)

and

\log x

through

(10, 1)

, so

\ln x

rises faster.

How can I tell if a log graph increases or decreases?

Look at the base. If

b > 1

, the graph increases; if

0 < b < 1

, it decreases. A minus sign in front, as in

y = -\log_2 x

, also turns an increasing graph into a decreasing one.

Glossary

Logarithmic function: A function $f(x) = \log_b x$ with $b > 0$ , $b \neq 1$ ; it gives the exponent to which $b$ must be raised to get $x$
Vertical asymptote: A vertical line $x = h$ that the graph gets closer and closer to without touching it; for $y = \log_b x$ it is $x = 0$
Domain: All x-values for which the function is defined; for $y = \log_b(x - h)$ it is $x > h$
Range: All y-values the function takes; for every logarithmic function it is all real numbers
Inverse function: A function that undoes another one; its graph is the reflection across $y = x$ , as $y = \log_b x$ is for $y = b^x$
Key points: Easy points to plot: $\left(\frac{1}{b}, -1\right)$ , $(1, 0)$ and $(b, 1)$ for $y = \log_b x$

Formula Card

Parent function

y = \log_b x \iff b^y = x

Base $b > 0$, $b \neq 1$; domain $x > 0$, range all real numbers

Key points

\left(\tfrac{1}{b}, -1\right),\ (1, 0),\ (b, 1)

Plot these three points first

Transformed function

y = \log_b(x - h) + k

Asymptote $x = h$, domain $x > h$, key points $(1 + h, k)$ and $(b + h, 1 + k)$

Reflections

y = -\log_b x,\quad y = \log_b(-x)

Across the $x$-axis and across the $y$-axis (domain $x < 0$)

Inverse of the exponential

(p, q) \text{ on } y = b^x \iff (q, p) \text{ on } y = \log_b x

Reflection across the line $y = x$

Teacher Guide: Graphing Logarithmic Functions

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Sound Level in Decibels

Binary Search in Computer Science

The pH Scale in Chemistry

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Parent function

Key points

Transformed function

Reflections

Inverse of the exponential

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Natural Logarithms (ln)