1.6.4 The natural logarithm function

What you'll learn

What the natural logarithm $ln⁡x\ln x$ means and how it is connected to $e^x$ .
How to recognise and sketch the graph of $y=ln⁡xy=\ln x$ .
How to use $ln⁡\ln$ to evaluate expressions and solve exponential equations.
How the logarithm laws apply to natural logarithms.

Prerequisite: logarithms as inverse operations

A logarithm tells you the power to which a base must be raised. For a positive base $a$ , where $a≠1a\neq 1$ ,

ay=x⟺log⁡ax=y.a^y=x \quad \Longleftrightarrow \quad \log_a x=y.

For example, $log_2 8=3$ because $2^3=8$ .

The word inverse means an operation or function that reverses another. Taking a logarithm to base $a$ reverses raising $a$ to a power.

Key Idea

Exponential and logarithmic forms

The statements $a^y=x$ and $log_a x=y$ contain exactly the same information. You can switch between the two forms whenever it is useful.

Example

Converting between logarithmic and exponential form

Express $log_5 125=3$ in exponential form, and express $e^4=k$ in logarithmic form.

The equation $log_5 125=3$ asks, “What power of 5 gives 125?” Therefore, its exponential form is $5^3=125$ .
In $e^4=k$ , the base is $e$ , the power is 4 and the result is $k$ .
Therefore, its logarithmic form is $log_e k=4$ , which is normally written as $ln⁡k=4\ln k=4$ .

The number $e$

The number $e$ is an important mathematical constant, approximately equal to 2.718. Like $π\pi$ , it is irrational, meaning that its decimal expansion continues forever without repeating.

The exponential function $y=e^x$ appears naturally in mathematical models involving continuous growth or decay. Its inverse function is the natural logarithm.

Definition

The natural logarithm

The natural logarithm of $x$ , written $ln⁡x\ln x$ , is the logarithm of $x$ to base $e$ :

ln x=\log_e x.

Equivalently, $ln⁡x\ln x$ is the power to which $e$ must be raised to obtain $x$ .

For example:

ln⁡1=0,ln⁡e=1,ln⁡(e5)=5.\ln 1=0,\qquad \ln e=1,\qquad \ln(e^5)=5.

These follow because $e^0=1$ , $e^1=e$ and $e^5=e^5$ .

Common Mistake

Reading the notation

The notation $ln⁡x\ln x$ means “the natural logarithm of $x$ ”. It does not mean $l$ multiplied by $n$ multiplied by $x$ .

Inverse relationships

Because $ln⁡x\ln x$ and $e^x$ are inverse functions, each one reverses the other:

ln(e^x)=x

for every real value of $x$ , and

e^{\ln x}=x

for $x > 0$ .

The condition $x > 0$ is essential because $ln⁡x\ln x$ is only defined for positive real inputs.

Example

Using inverse functions

Simplify $ln(e^{3t-2})$ and $e^{\ln 7}$ .

In $ln(e^{3t-2})$ , the natural logarithm directly reverses the exponential function.
Therefore, $ln(e^{3t-2})=3t-2$ .
Similarly, the exponential function reverses the natural logarithm, so $e^{\ln 7}=7$ .

Common Mistake

Order matters

The identity $ln(e^x)=x$ does not mean that $ln x=e^x$ . The functions undo one another only when one is applied to the output of the other.

The graph of $y=ln⁡xy=\ln x$

The graph of $y=ln⁡xy=\ln x$ is the reflection of the graph of $y=e^x$ in the line $y = x$ . This is a general property of the graphs of inverse functions: their input and output coordinates are exchanged.

For example, $y=e^x$ passes through $(0, 1)$ , so $y=ln⁡xy=\ln x$ passes through $(1, 0)$ .

Graph of y = ln x and y = e^x reflected in y = x, with their key points and the asymptote x = 0 labelled

Domain and range

The domain of a function is the set of permitted input values. Since no real power of $e$ produces zero or a negative number, the domain of $y=ln⁡xy=\ln x$ is

x > 0.

The range is the set of possible output values. Every real number can be an output of $ln⁡x\ln x$ , so its range is

−∞<y<∞.-\infty<y<\infty.

Intercept and asymptote

The graph crosses the $x$ -axis at $(1, 0)$ because $ln⁡1=0\ln 1=0$ .

It has no $y$ -intercept because $x = 0$ is not in its domain.

The line $x = 0$ , which is the $y$ -axis, is a vertical asymptote. An asymptote is a line that a curve approaches increasingly closely without meeting. In particular,

x→0+.\ln x\to-\infty \quad \text{as } x\to 0^+.

The notation $x→0+x\to 0^+$ means that $x$ approaches zero through positive values.

Shape of the curve

The function $ln⁡x\ln x$ is increasing throughout its domain: if $x$ increases, then $ln⁡x\ln x$ also increases. However, the graph becomes gradually less steep, so its rate of increase slows.

For large positive $x$ , $ln⁡x\ln x$ still increases, but very slowly. There is no horizontal asymptote.

Key Idea

Features of the logarithm graph

For $y=ln⁡xy=\ln x$ :

the domain is $x > 0$ ;
the range is all real numbers;
the $x$ -intercept is $(1, 0)$ ;
there is no $y$ -intercept;
$x = 0$ is a vertical asymptote;
the curve is increasing but becomes less steep.

Example

Sketching a natural logarithm graph

Sketch $y=ln⁡xy=\ln x$ and mark its main features.

Draw the vertical asymptote $x = 0$ and keep the curve entirely to its right because the domain is $x > 0$ .
Mark the exact point $(1, 0)$ , where the graph crosses the $x$ -axis.
Draw an increasing curve that falls towards negative infinity as $x$ approaches zero from the right, then rises increasingly slowly as $x$ becomes large.

Evaluating natural logarithms

Some natural logarithms have exact values, especially when the input is a power of $e$ :

ln(e^k)=k.

For other positive inputs, use the $ln⁡\ln$ button on your calculator. For example, $ln⁡10≈2.303\ln 10\approx 2.303$ to three decimal places.

A calculator should be in the mode requested by the wider question, but degree or radian mode does not affect a direct logarithm calculation.

Example

Evaluating a natural logarithm

Evaluate $ln⁡15\ln 15$ to three decimal places.

Check that the input is positive, so the natural logarithm is defined.
Enter $ln⁡(15)\ln(15)$ into the calculator to obtain approximately 2.708050201.
Round to three decimal places: $ln⁡15≈2.708\ln 15\approx 2.708$ .

Natural logarithm laws

Because $ln⁡x\ln x$ is a logarithm, the standard logarithm laws apply. For positive $a$ and $b$ ,

ln⁡(ab)=ln⁡a+ln⁡b,\ln(ab)=\ln a+\ln b,

ln⁡(ab)=ln⁡a−ln⁡b,\ln\left(\frac{a}{b}\right)=\ln a-\ln b,

and, for any real number $k$ ,

ln(a^k)=k\ln a.

Common Mistake

Logarithms do not distribute over addition

In general, $ln⁡(a+b)≠ln⁡a+ln⁡b\ln(a+b)\neq\ln a+\ln b$ . The addition law applies when the arguments are multiplied: $ln⁡(ab)=ln⁡a+ln⁡b\ln(ab)=\ln a+\ln b$ .

Example

Simplifying a logarithmic expression

Simplify $ln(3x^2)-\ln 3$ , where $x≠0x\neq 0$ .

Apply the quotient law:
$ln⁡(3x2)−ln⁡3=ln⁡(3x23).\ln(3x^2)-\ln 3=\ln\left(\frac{3x^2}{3}\right).$
Simplify the argument to obtain $ln(x^2)$ .
Since $x^2>0$ when $x≠0x\neq 0$ , the logarithm is defined. The simplified expression is therefore $ln(x^2)$ .

Solving equations using $ln⁡\ln$

Natural logarithms are especially useful when the unknown appears in the power of $e$ . Taking $ln⁡\ln$ of both sides allows you to bring the exponent down.

Example

Solving an exponential equation

Solve $e^{2x-1}=7$ , giving your answer to three decimal places.

Take the natural logarithm of both sides:
$ln(e^{2x-1})=\ln 7.$
Use $ln(e^u)=u$ to obtain $2x−1=ln⁡72x-1=\ln 7$ .
Rearrange:
$x=1+ln⁡72≈1.473.x=\frac{1+\ln 7}{2}\approx 1.473.$

Tip

Check solutions graphically

The solution of $ln⁡x=k\ln x=k$ is the $x$ -coordinate where the horizontal line $y = k$ meets $y=ln⁡xy=\ln x$ . Since the logarithm graph is always increasing, there is exactly one such solution.

Exam technique

In the exam

For a graph sketch, show the asymptote $x = 0$ , the intercept $(1, 0)$ and the correct increasing shape.
Check that every input to $ln⁡\ln$ is positive; reject values that make a logarithm’s argument zero or negative.
Use exact forms such as $ln⁡7\ln 7$ unless the question asks for a decimal, and only round at the end.

Self review

Check yourself

What are the domain, range, intercept and asymptote of $y=ln⁡xy=\ln x$ ?
How would you solve $e^{3x}=11$ using natural logarithms?
Why is $ln⁡(−2)\ln(-2)$ not defined as a real number?

What you'll learn

What the natural logarithm $ln⁡x\ln x$ means and how it is connected to $e^x$ .
How to recognise and sketch the graph of $y=ln⁡xy=\ln x$ .
How to use $ln⁡\ln$ to evaluate expressions and solve exponential equations.
How the logarithm laws apply to natural logarithms.

Prerequisite: logarithms as inverse operations

A logarithm tells you the power to which a base must be raised. For a positive base $a$ , where $a≠1a\neq 1$ ,

ay=x⟺log⁡ax=y.a^y=x \quad \Longleftrightarrow \quad \log_a x=y.

For example, $log_2 8=3$ because $2^3=8$ .

The word inverse means an operation or function that reverses another. Taking a logarithm to base $a$ reverses raising $a$ to a power.

Key Idea

Exponential and logarithmic forms

The statements $a^y=x$ and $log_a x=y$ contain exactly the same information. You can switch between the two forms whenever it is useful.

Example

Converting between logarithmic and exponential form

Express $log_5 125=3$ in exponential form, and express $e^4=k$ in logarithmic form.

The equation $log_5 125=3$ asks, “What power of 5 gives 125?” Therefore, its exponential form is $5^3=125$ .
In $e^4=k$ , the base is $e$ , the power is 4 and the result is $k$ .
Therefore, its logarithmic form is $log_e k=4$ , which is normally written as $ln⁡k=4\ln k=4$ .

The number $e$

The number $e$ is an important mathematical constant, approximately equal to 2.718. Like $π\pi$ , it is irrational, meaning that its decimal expansion continues forever without repeating.

The exponential function $y=e^x$ appears naturally in mathematical models involving continuous growth or decay. Its inverse function is the natural logarithm.

Definition

The natural logarithm

The natural logarithm of $x$ , written $ln⁡x\ln x$ , is the logarithm of $x$ to base $e$ :

ln x=\log_e x.

Equivalently, $ln⁡x\ln x$ is the power to which $e$ must be raised to obtain $x$ .

For example:

ln⁡1=0,ln⁡e=1,ln⁡(e5)=5.\ln 1=0,\qquad \ln e=1,\qquad \ln(e^5)=5.

These follow because $e^0=1$ , $e^1=e$ and $e^5=e^5$ .

Common Mistake

Reading the notation

The notation $ln⁡x\ln x$ means “the natural logarithm of $x$ ”. It does not mean $l$ multiplied by $n$ multiplied by $x$ .

Inverse relationships

Because $ln⁡x\ln x$ and $e^x$ are inverse functions, each one reverses the other:

ln(e^x)=x

for every real value of $x$ , and

e^{\ln x}=x

for $x > 0$ .

The condition $x > 0$ is essential because $ln⁡x\ln x$ is only defined for positive real inputs.

Example

Using inverse functions

Simplify $ln(e^{3t-2})$ and $e^{\ln 7}$ .

In $ln(e^{3t-2})$ , the natural logarithm directly reverses the exponential function.
Therefore, $ln(e^{3t-2})=3t-2$ .
Similarly, the exponential function reverses the natural logarithm, so $e^{\ln 7}=7$ .

Common Mistake

Order matters

The identity $ln(e^x)=x$ does not mean that $ln x=e^x$ . The functions undo one another only when one is applied to the output of the other.

The graph of $y=ln⁡xy=\ln x$

For example, $y=e^x$ passes through $(0, 1)$ , so $y=ln⁡xy=\ln x$ passes through $(1, 0)$ .

Graph of y = ln x and y = e^x reflected in y = x, with their key points and the asymptote x = 0 labelled

Domain and range

The domain of a function is the set of permitted input values. Since no real power of $e$ produces zero or a negative number, the domain of $y=ln⁡xy=\ln x$ is

x > 0.

The range is the set of possible output values. Every real number can be an output of $ln⁡x\ln x$ , so its range is

−∞<y<∞.-\infty<y<\infty.

Intercept and asymptote

The graph crosses the $x$ -axis at $(1, 0)$ because $ln⁡1=0\ln 1=0$ .

It has no $y$ -intercept because $x = 0$ is not in its domain.

The line $x = 0$ , which is the $y$ -axis, is a vertical asymptote. An asymptote is a line that a curve approaches increasingly closely without meeting. In particular,

x→0+.\ln x\to-\infty \quad \text{as } x\to 0^+.

The notation $x→0+x\to 0^+$ means that $x$ approaches zero through positive values.

Shape of the curve

For large positive $x$ , $ln⁡x\ln x$ still increases, but very slowly. There is no horizontal asymptote.

Key Idea

Features of the logarithm graph

For $y=ln⁡xy=\ln x$ :

the domain is $x > 0$ ;
the range is all real numbers;
the $x$ -intercept is $(1, 0)$ ;
there is no $y$ -intercept;
$x = 0$ is a vertical asymptote;
the curve is increasing but becomes less steep.

Example

Sketching a natural logarithm graph

Sketch $y=ln⁡xy=\ln x$ and mark its main features.

Draw the vertical asymptote $x = 0$ and keep the curve entirely to its right because the domain is $x > 0$ .
Mark the exact point $(1, 0)$ , where the graph crosses the $x$ -axis.
Draw an increasing curve that falls towards negative infinity as $x$ approaches zero from the right, then rises increasingly slowly as $x$ becomes large.

Evaluating natural logarithms

Some natural logarithms have exact values, especially when the input is a power of $e$ :

ln(e^k)=k.

For other positive inputs, use the $ln⁡\ln$ button on your calculator. For example, $ln⁡10≈2.303\ln 10\approx 2.303$ to three decimal places.

A calculator should be in the mode requested by the wider question, but degree or radian mode does not affect a direct logarithm calculation.

Example

Evaluating a natural logarithm

Evaluate $ln⁡15\ln 15$ to three decimal places.

Check that the input is positive, so the natural logarithm is defined.
Enter $ln⁡(15)\ln(15)$ into the calculator to obtain approximately 2.708050201.
Round to three decimal places: $ln⁡15≈2.708\ln 15\approx 2.708$ .

Natural logarithm laws

Because $ln⁡x\ln x$ is a logarithm, the standard logarithm laws apply. For positive $a$ and $b$ ,

ln⁡(ab)=ln⁡a+ln⁡b,\ln(ab)=\ln a+\ln b,

ln⁡(ab)=ln⁡a−ln⁡b,\ln\left(\frac{a}{b}\right)=\ln a-\ln b,

and, for any real number $k$ ,

ln(a^k)=k\ln a.

Common Mistake

Logarithms do not distribute over addition

In general, $ln⁡(a+b)≠ln⁡a+ln⁡b\ln(a+b)\neq\ln a+\ln b$ . The addition law applies when the arguments are multiplied: $ln⁡(ab)=ln⁡a+ln⁡b\ln(ab)=\ln a+\ln b$ .

Example

Simplifying a logarithmic expression

Simplify $ln(3x^2)-\ln 3$ , where $x≠0x\neq 0$ .

Apply the quotient law:
$ln⁡(3x2)−ln⁡3=ln⁡(3x23).\ln(3x^2)-\ln 3=\ln\left(\frac{3x^2}{3}\right).$
Simplify the argument to obtain $ln(x^2)$ .
Since $x^2>0$ when $x≠0x\neq 0$ , the logarithm is defined. The simplified expression is therefore $ln(x^2)$ .

Solving equations using $ln⁡\ln$

Natural logarithms are especially useful when the unknown appears in the power of $e$ . Taking $ln⁡\ln$ of both sides allows you to bring the exponent down.

Example

Solving an exponential equation

Solve $e^{2x-1}=7$ , giving your answer to three decimal places.

Take the natural logarithm of both sides:
$ln(e^{2x-1})=\ln 7.$
Use $ln(e^u)=u$ to obtain $2x−1=ln⁡72x-1=\ln 7$ .
Rearrange:
$x=1+ln⁡72≈1.473.x=\frac{1+\ln 7}{2}\approx 1.473.$

Tip

Check solutions graphically

Exam technique

In the exam

For a graph sketch, show the asymptote $x = 0$ , the intercept $(1, 0)$ and the correct increasing shape.
Check that every input to $ln⁡\ln$ is positive; reject values that make a logarithm’s argument zero or negative.
Use exact forms such as $ln⁡7\ln 7$ unless the question asks for a decimal, and only round at the end.

Self review

Check yourself

What are the domain, range, intercept and asymptote of $y=ln⁡xy=\ln x$ ?
How would you solve $e^{3x}=11$ using natural logarithms?
Why is $ln⁡(−2)\ln(-2)$ not defined as a real number?

What you'll learn

Prerequisite: logarithms as inverse operations

The number $e$

Inverse relationships

The graph of $y=ln⁡xy=\ln x$

Domain and range

Intercept and asymptote

Shape of the curve

Evaluating natural logarithms

Natural logarithm laws

Solving equations using $ln⁡\ln$

Review 1.6.4 The natural logarithm function by teaching Genie

1.6.4 The natural logarithm function

What you'll learn

Prerequisite: logarithms as inverse operations

The number $e$

Inverse relationships

The graph of $y=ln⁡xy=\ln x$

Domain and range

Intercept and asymptote

Shape of the curve

Evaluating natural logarithms

Natural logarithm laws

Solving equations using $ln⁡\ln$

Review 1.6.4 The natural logarithm function by teaching Genie

1.6.4 The natural logarithm function

What you'll learn

Prerequisite: logarithms as inverse operations

The number eee

Inverse relationships

The graph of y=ln⁡xy=\ln xy=lnx

Domain and range

Intercept and asymptote

Shape of the curve

Evaluating natural logarithms

Natural logarithm laws

Solving equations using ln⁡\lnln

Review 1.6.4 The natural logarithm function by teaching Genie

1.6.4 The natural logarithm function Revision Guide

1.6.4 The natural logarithm function

What you'll learn

Prerequisite: logarithms as inverse operations

The number eee

Inverse relationships

The graph of y=ln⁡xy=\ln xy=lnx

Domain and range

Intercept and asymptote

Shape of the curve

Evaluating natural logarithms

Natural logarithm laws

Solving equations using ln⁡\lnln

Review 1.6.4 The natural logarithm function by teaching Genie

1.6.4 The natural logarithm function Revision Guide

The number $e$

The graph of $y=ln⁡xy=\ln x$

Solving equations using $ln⁡\ln$

The number $e$

The graph of $y=ln⁡xy=\ln x$

Solving equations using $ln⁡\ln$