Logarithms and the Natural Logarithm

Understanding logarithms as inverses of exponentials, with focus on ln x

SPEC 6.3 • EXPONENTIALS AND LOGARITHMS

What is a Logarithm?

A logarithm answers the question: "What power do I need?"

Definition

lo g_{a} b = x means a^{x} = b

"The logarithm base a of b equals x" means "a to the power x equals b"

Reading Logarithms

lo g_{2} 8 = 3

"What power of 2 gives 8?"

Answer: 3 (because

2^{3} = 8

)

The Inverse

Logarithms are the inverse of exponentials.

If

a^{x} = b

, then

x = lo g_{a} b

Converting Between Forms

Key Relationship

a^{x} = b ⟺ lo g_{a} b = x

Exponential FormLogarithmic Form

2^{3} = 8

lo g_{2} 8 = 3

1 0^{2} = 100

lo g_{10} 100 = 2

5^{4} = 625

lo g_{5} 625 = 4

e^{1} = e

ln e = 1

3^{- 2} = \frac{1}{9}

lo g_{3} \frac{1}{9} = - 2

Tip

The base stays the same, the exponent becomes the answer, and the result becomes the input.

Key Logarithm Values

These results are true for any base a > 0, a ≠ 1:

Result 1

lo g_{a} 1 = 0

Because

a^{0} = 1

Result 2

lo g_{a} a = 1

Because

a^{1} = a

Result 3

lo g_{a} a^{n} = n

The logarithm "cancels" the exponential with the same base

Inverse Property

a^{l o g_{a} x} = x

The exponential and logarithm are inverses — they undo each other.

Common Logarithm (log₁₀)

Definition

The common logarithm is logarithm base 10:

lo g_{10} x = lo g x

When no base is written, it means base 10.

ExpressionValueBecause...

lo g 10

1

1 0^{1} = 10

lo g 100

2

1 0^{2} = 100

lo g 1000

3

1 0^{3} = 1000

lo g 1

0

1 0^{0} = 1

lo g 0.1

−1

1 0^{- 1} = 0.1

Calculator

Press the log button for common logarithm (base 10).

Calculator: Logarithm with Any Base

To evaluate

lo g_{a} b

on your calculator, you have two methods:

Method 1: Direct Entry (Casio)

Most scientific calculators have a

lo g_{□} □

function:

SHIFT → log → enter base → enter number

Example: For

lo g_{2} 8

, press: SHIFT log 2 , 8 = → gives 3

Method 2: Change of Base Formula

lo g_{a} b = \frac{l n b}{l n a} = \frac{l o g b}{l o g a}

Convert any logarithm to a fraction using ln or log.

Example

Find

lo g_{5} 20

using the change of base formula:

lo g_{5} 20 = \frac{l n 20}{l n 5} = \frac{2.996...}{1.609...} = 1.861

(3 s.f.)

Tip

You can use either ln or log in the formula — both give the same answer!

The Natural Logarithm (ln)

Definition

The natural logarithm is logarithm base e:

lo g_{e} x = ln x

Written as "ln" (pronounced "ell-en" or "natural log")

ln and e are Inverses

The functions

e^{x}

and

ln x

are inverse functions — they undo each other:

e^{l n x} = x and ln (e^{x}) = x

This is the most important relationship to remember!

Calculator

Press ln for natural logarithm.

Press SHIFT → ln to get

e^{x}

.

Important

ln x \neq = lo g x

ln uses base e ≈ 2.718

log uses base 10

Key Values of ln

These results follow directly from the definition

ln x = lo g_{e} x

:

ResultValueBecause...

ln 1

0

e^{0} = 1

ln e

1

e^{1} = e

ln e^{n}

n

Logarithm undoes exponential

e^{l n x}

x

Exponential undoes logarithm

The Inverse Pair

ln (e^{x}) = x and e^{l n x} = x

These are the most important results — memorise them!

Domain

ln x is only defined for x > 0. You cannot take ln of zero or negative numbers.

01Worked Example

Convert to logarithmic form:

(a)

2^{5} = 32

(b)

e^{3} \approx 20.09

(c)

1 0^{- 2} = 0.01

Use:

a^{x} = b

becomes

lo g_{a} b = x

(a)

2^{5} = 32

→

lo g_{2} 32 = 5

(b)

e^{3} \approx 20.09

→

ln 20.09 = 3

(or

lo g_{e} 20.09 = 3

)

(c)

1 0^{- 2} = 0.01

→

lo g 0.01 = - 2

02Worked Example

Convert to exponential form:

(a)

lo g_{2} 8 = 3

(b)

ln 7 \approx 1.95

(c)

lo g 1000 = 3

Use:

lo g_{a} b = x

becomes

a^{x} = b

(a)

lo g_{2} 8 = 3

→

2^{3} = 8

(b)

ln 7 \approx 1.95

→

e^{1.95} \approx 7

(c)

lo g 1000 = 3

→

1 0^{3} = 1000

03Worked Example

Evaluate without a calculator:

(a)

lo g_{2} 16

(b)

lo g_{3} 27

(c)

lo g_{5} \frac{1}{25}

Ask: "What power of the base gives this number?"

(a)

lo g_{2} 16 = ?

→

2^{?} = 16

→

2^{4} = 16

→ Answer: 4

(b)

lo g_{3} 27 = ?

→

3^{?} = 27

→

3^{3} = 27

→ Answer: 3

(c)

lo g_{5} \frac{1}{25} = ?

→

5^{?} = \frac{1}{25}

→

5^{- 2} = \frac{1}{25}

→ Answer: −2

04Worked Example

Evaluate:

(a)

ln e^{4}

(b)

e^{l n 5}

(c)

ln e + ln 1

(d)

ln e

Use:

ln e^{n} = n

and

e^{l n x} = x

(a)

ln e^{4} = 4

(ln and e cancel, leaving the power)

(b)

e^{l n 5} = 5

(e and ln cancel, leaving the argument)

(c)

ln e + ln 1 = 1 + 0 = 1

(d)

ln e = ln e^{1/2} = \frac{1}{2}

05Worked Example

Solve for x:

(a)

ln x = 2

(b)

e^{x} = 7

(c)

lo g x = 3

To undo ln, use e^. To undo e^, use ln. To undo log, use 10^.

(a)

ln x = 2

→

x = e^{2} \approx 7.39

(b)

e^{x} = 7

→

x = ln 7 \approx 1.95

(c)

lo g x = 3

→

x = 1 0^{3} = 1000

Graph of y = ln x

Key Features

Domainx > 0RangeAll real numbersAsymptotex = 0 (vertical)Intercept(1, 0)Key point(e, 1) ≈ (2.72, 1)

Remember

ln 1 = 0 → curve passes through (1, 0)

ln e = 1 → curve passes through (e, 1)

Domain

ln x only exists for positive x values. The curve never touches or crosses the y-axis.

Inverse Relationship

The graphs of

y = e^{x}

and

y = ln x

are reflections of each other in the line

y = x

.

Inverse Functions

e^{l n x} = x and ln (e^{x}) = x

Point (a, b) on

y = e^{x}

↔ Point (b, a) on

y = ln x

Practice

Question 1:

(a) Convert

3^{4} = 81

to logarithmic form.

(b) Convert

lo g_{2} 32 = 5

to exponential form.

(a)

3^{4} = 81

→

lo g_{3} 81 = 4

(b)

lo g_{2} 32 = 5

→

2^{5} = 32

Practice

Question 2: Evaluate without a calculator:

(a)

lo g_{4} 64

(b)

ln e^{3}

(c)

lo g 10000

(a)

lo g_{4} 64 = ?

→

4^{?} = 64

→

4^{3} = 64

→ Answer: 3

(b)

ln e^{3} = 3

(using

ln e^{n} = n

)

(c)

lo g 10000 = lo g 1 0^{4} = 4

Practice

Question 3: Sketch the graph of

y = ln (x - 1)

, stating:

(a) The equation of the asymptote

(b) The coordinates of any axis intercepts

y = ln (x - 1)

is

y = ln x

translated 1 unit right.

(a)

Asymptote: Original is x = 0, so new asymptote is x = 1

(b)

X-intercept: Set y = 0 →

ln (x - 1) = 0

→

x - 1 = 1

→

x = 2

X-intercept: (2, 0)

No y-intercept (asymptote at x = 1, so x = 0 is not in domain)

Summary

Key Definition

lo g_{a} b = x \Leftrightarrow a^{x} = b

Natural Log

ln x = lo g_{e} x

ResultExplanation

lo g_{a} 1 = 0

Because

a^{0} = 1

lo g_{a} a = 1

Because

a^{1} = a

ln e^{n} = n

ln and e cancel

e^{l n x} = x

e and ln cancel

Graph of y = ln x

Domain: x > 0 | Range: ℝ | Through (1, 0) and (e, 1) | Asymptote: x = 0 | Reflection of y = eˣ in y = x