Teacher Guide: Natural Logarithms (ln)

Learn what ln x means, why ln e = 1 and ln 1 = 0, how ln undoes e^x, and how to solve equations like e^x = 20 with a calculator.

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Learning Objectives

Define the natural logarithm as the logarithm with base $e$
Convert between $\ln x = y$ and $e^y = x$
Use $\ln 1 = 0$ , $\ln e = 1$ , $\ln(e^x) = x$ and $e^{\ln x} = x$ to simplify expressions
Evaluate natural logarithms with a calculator and check the result
Solve simple exponential equations of the form $ae^{kx} = b$ using $\ln$

Prerequisites

• The Natural Base e: the number $e \approx 2.71828$ and the function $e^x$
• Continuous Growth and Decay: models of the form $A = Pe^{rt}$
• Definition of a Logarithm: $\log_b y = x$ means $b^x = y$
• Common Logarithms (Base 10): using the LOG key on a calculator

Discussion Starters

1. Your calculator has two log keys, LOG and LN. Why do you think mathematicians need both?
2. Without a calculator, between which two whole numbers is $\ln 10$ ? How do you know?
3. Why is $\ln x$ negative when $x$ is between 0 and 1?
4. If $e^x$ and $\ln x$ undo each other, what should the graph of one look like compared to the other?

Common Misconceptions

$\ln$ is a number that multiplies $x$ (treating $\ln x$ as $\ln \times x$ )

Remediation: Stress that $\ln$ is a function, like $\sqrt{\phantom{x}}$ . Always read $\ln x$ as a question: " $e$ to what power gives $x$ ?" Write $\ln(x)$ with brackets at first.

$\ln$ of a small number cannot be negative

Remediation: Show the table: $\ln 0.5 \approx -0.693$ because $e^{-0.693} \approx 0.5$ . Negative exponents give results between 0 and 1, so the ln of such numbers is negative.

$e^x = k$ is solved by dividing $k$ by $e$

Remediation: Compare with $x^2 = 9$ : we undo a square with a square root, not by dividing by 2. In the same way, we undo $e^x$ with $\ln$ .

Differentiation Ideas

For Struggling Students:

• Start from a table of powers of $e$ ( $e^0 = 1$ , $e^1 \approx 2.718$ , $e^2 \approx 7.389$ ) and read ln values backwards
• Practise only the inverse rules $\ln(e^x) = x$ and $e^{\ln x} = x$ before using a calculator
• Use a two-column chart: exponential form and ln form

For On-Level Students:

• Evaluate ln values with a calculator and check each with $e^x$
• Solve equations of the form $ae^{kx} = b$
• Apply $\ln$ to continuous growth and decay contexts

For Advanced Students:

• Explain the rule of 70 for doubling time using $\ln 2 \approx 0.693$
• Compare $\ln x$ and $\log x$ for the same inputs and find the constant ratio between them
• Investigate why $\ln x$ grows so slowly: how large must $x$ be for $\ln x = 10$ ?

Standards Alignment

HSF-LE.A.4 (CCSS.MATH.CONTENT.HSF.LE.A.4)
Express the solution to exponential models as a logarithm (base 2, 10 or e) and evaluate it using technology
HSF-BF.B.5 (CCSS.MATH.CONTENT.HSF.BF.B.5)
Understand the inverse relationship between exponents and logarithms
HSF-BF.B.4 (CCSS.MATH.CONTENT.HSF.BF.B.4)
Find inverse functions

Lesson Resources

visualMirror Graphs
Interactive graph of $y = e^x$ and $y = \ln x$ reflected in the line $y = x$ , with matching points
activityInverse Pairs Card Sort
Match cards like $\ln(e^4)$ , $e^{\ln 9}$ , $\ln 1$ and $\ln e$ with their values
worksheetWhen Does It Happen?
Continuous growth and decay problems solved with $\ln$ : bacteria, savings, medicine

Lesson Content

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

Definition

The natural logarithm is the logarithm with base

e

, where

e \approx 2.71828

is Euler's number. Instead of writing

\log_e x

, we write

\ln x

(read "el-en of x" or "natural log of x").

\ln x = \log_e x

Like every logarithm,

\ln x

answers a question: **"To which power must I raise

e

to get

x

?"**

\ln x = y \iff e^y = x

Two values to know by heart:

$\ln 1 = 0$ , because $e^0 = 1$
$\ln e = 1$ , because $e^1 = e$

ln undoes e^x (and the other way round): the function

y = \ln x

is the inverse function of

y = e^x

, so

\ln(e^x) = x \quad \text{and} \quad e^{\ln x} = x \; (x > 0)

The argument must be positive:

\ln x

only exists for

x > 0

, because

e^y

is never zero or negative. So the domain of

y = \ln x

x > 0

Worked Examples

Evaluate $\ln(e^5)$ without a calculator.

Rewrite as a question

$\ln(e^5)$ asks: to which power must I raise $e$ to get $e^5$ ? → $e^{?} = e^5$

Read off the exponent

The power that turns $e$ into $e^5$ is $5$ → $? = 5$

Use the inverse rule

$\ln$ undoes $e^x$ : $\ln(e^x) = x$ → $\ln(e^5) = 5$

Answer: $\ln(e^5) = 5$

Common Mistakes

Writing $\ln 1 = 1$ or $\ln e = 0$

Why it's wrong: The two key values are easy to swap. $\ln$ gives an exponent, not the number itself.

Correct: $\ln 1 = 0$ because $e^0 = 1$ , and $\ln e = 1$ because $e^1 = e$ .

Taking $\ln$ of zero or a negative number, like $\ln 0$ or $\ln(-3)$

Why it's wrong: $e^y$ is always positive, whatever $y$ is. No power of $e$ gives $0$ or a negative number.

Correct: $\ln x$ only exists for $x > 0$ . The calculator shows an error for $\ln 0$ and $\ln(-3)$ .

Pressing LOG instead of LN on the calculator

Why it's wrong: LOG is the common logarithm with base 10, LN is the natural logarithm with base $e$ . They give different results.

Correct: $\log 10 = 1$ , but $\ln 10 \approx 2.303$ . Use LN whenever the base is $e$ .

Solving $e^x = 20$ by dividing: $x = \frac{20}{e}$

Why it's wrong: $x$ is an exponent, not a factor. Dividing by $e$ does not undo raising $e$ to a power.

Correct: Undo $e^x$ with $\ln$ : $x = \ln 20 \approx 2.996$ .

Why It Matters

The number

e

appears whenever something grows or shrinks continuously, and

\ln

is the tool that brings the time or the rate back out of the exponent:

Continuous growth: if bacteria grow as $N = 500e^{0.4t}$ , then $\ln$ tells you when there will be 2000 of them.
Money: with continuous compounding $A = Pe^{rt}$ , the doubling time is $t = \frac{\ln 2}{r}$ .
Medicine: how long a drug stays above a safe level in the blood is found with $\ln$ .
Science and calculus: $\ln x$ shows up in physics, chemistry and every calculus course, because $e^x$ and $\ln x$ have the simplest derivatives.

Without

\ln

, an equation like

e^x = 20

could only be solved by guessing. With

\ln

, it takes one step:

x = \ln 20 \approx 2.996

Real World Applications

Biology: When Will the Culture Reach a Target?

A bacteria culture with plenty of food grows continuously as $N = N_0 e^{rt}$. To find WHEN it reaches a size, you need $\ln$ to bring $t$ down from the exponent.

Example:

With $N = 1000e^{0.3t}$ , the culture reaches 3000 when $e^{0.3t} = 3$ , so $0.3t = \ln 3$ and $t = \frac{\ln 3}{0.3} \approx 3.66$ hours.

1Try It Yourself

A culture starts with 500 bacteria and grows as $N = 500e^{0.4t}$ , where $t$ is in hours.

After how many hours are there 2000 bacteria? Round to 2 decimal places.

Step 1: Write the mathematical expression

Calculate $t = \frac{\ln 4}{0.4}$ :

Finance: Doubling Time with Continuous Interest

Banks compare savings offers with continuous compounding, $A = Pe^{rt}$. The time it takes to double money is $t = \frac{\ln 2}{r}$.

Example:

At 7% compounded continuously, money doubles after $\frac{\ln 2}{0.07} \approx 9.90$ years. Since $\ln 2 \approx 0.693$ , bankers use the "rule of 70": 70 divided by the rate in percent.

2Try It Yourself

You invest 1000 dollars at 5% interest compounded continuously, so $A = 1000e^{0.05t}$ .

After how many years will you have 2000 dollars? Round to 2 decimal places.

Step 1: Write the mathematical expression

Calculate $t = \frac{\ln 2}{0.05}$ :

Medicine: How Long a Drug Stays in the Blood

Many medicines leave the body continuously, so the amount in the blood follows $C = C_0 e^{-kt}$. Doctors use $\ln$ to work out when the next dose is due.

Example:

A drug with $k = 0.25$ per hour has a half-life of $\frac{\ln 2}{0.25} \approx 2.77$ hours: after that time, half of it is left.

3Try It Yourself

A patient takes 100 mg of a medicine. The amount left after $t$ hours is $C = 100e^{-0.25t}$ .

After how many hours are 40 mg left? Round to 2 decimal places.

Step 1: Write the mathematical expression

Calculate $t = \frac{\ln 0.4}{-0.25}$ :

Key Takeaways

1The natural logarithm is the logarithm with base $e \approx 2.71828$ : $\ln x = \log_e x$
2 $\ln x = y$ means $e^y = x$ : ln gives the power of $e$ that produces $x$
3Key values: $\ln 1 = 0$ and $\ln e = 1$
4 $\ln$ and $e^x$ are inverse functions: $\ln(e^x) = x$ and $e^{\ln x} = x$ for $x > 0$
5 $\ln x$ only exists for $x > 0$ ; it is negative for $0 < x < 1$ and positive for $x > 1$
6To solve $e^x = k$ (with $k > 0$ ), take ln of both sides: $x = \ln k$

Frequently Asked Questions

Why is it called the natural logarithm?

Because its base

e

comes up naturally: in continuous growth, in continuous compounding and in calculus, where

e^x

and

\ln x

have the simplest rates of change. The "n" in

\ln

stands for the Latin "logarithmus naturalis".

Is ln the same as log?

No.

\log x

(common logarithm) has base 10 and

\ln x

has base

e

. For example,

\log 100 = 2

but

\ln 100 \approx 4.605

. Both answer the same kind of question, only the base is different.

Why does my calculator give an error for ln(0)?

Because

e^y

is always positive: there is no power of

e

that equals

0

or a negative number. The graph of

y = \ln x

only lives to the right of the

y

-axis, which is a vertical asymptote.

How are the graphs of $y = e^x$ and $y = \ln x$ related?

They are mirror images of each other in the line

y = x

, because they are inverse functions. The point

(0, 1)

y = e^x

matches the point

(1, 0)

y = \ln x

, and

(1, e)

matches

(e, 1)

Glossary

Natural logarithm: The logarithm with base $e$ , written $\ln x$ ; $\ln x = y$ means $e^y = x$
Euler's number $e$: The irrational number $e \approx 2.71828$ , the base of continuous growth
Inverse function: A function that undoes another one; $\ln x$ and $e^x$ are inverse functions
Argument: The number you take the logarithm of, the $x$ in $\ln x$ ; it must be positive
Domain: All allowed inputs of a function; for $y = \ln x$ the domain is $x > 0$
Exponential equation: An equation with the unknown in the exponent, such as $e^x = 20$

Formula Card

Definition

\ln x = \log_e x

Logarithm with base $e \approx 2.71828$

Exponential form

\ln x = y \iff e^y = x

Switch between the two forms

Key values

\ln 1 = 0, \quad \ln e = 1

Because $e^0 = 1$ and $e^1 = e$

Inverse rules

\ln(e^x) = x, \quad e^{\ln x} = x

The second one for $x > 0$

Solving $e^x = k$

x = \ln k

Only for $k > 0$

Teacher Guide: Natural Logarithms (ln)

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

Biology: When Will the Culture Reach a Target?

Finance: Doubling Time with Continuous Interest

Medicine: How Long a Drug Stays in the Blood

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Definition

Exponential form

Key values

Inverse rules

Solving $e^x = k$

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Graphing Logarithmic Functions