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 $(0, 1)$ on $y = e^x$ matches the point $(1, 0)$ on $y = \ln x$, and $(1, e)$ matches $(e, 1)$.

Teacher Resources

Natural Logarithms (ln)

Q: 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".

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.

Advanced25 minLesson

Printable worksheet

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

Try it now

What is the value of

\ln 1

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 $\ln$ : $x = \ln 20 \approx 2.996$ .

Interactive Visual

Function graph

Tap or hover over the graph to read off values.

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y = xAsymptote

Table of values

x	0.5	1	2	3	5
e^x	1.64872	2.71828	7.38906	20.0855	148.413
ln(x)	-0.693147	0	0.693147	1.09861	1.60944

Interactive Sandbox

Interactive Grapher

Tap or hover over the graph to read off values.

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

What is the value of $\ln 1$ ?

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

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".

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 $k > 0$

Natural Logarithms (ln)

Definition

Worked Examples

Common Mistakes

Interactive Visual

Function graph

Table of values

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

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