Sketch the graph of y3 log

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Logarithmic graphs

The graph of

y

=

log

₁₀

x

The graph of y= log₁₀x can be established by first completing a table of values, correct to 2 decimal places, by using your calculator to obtain the y-values.

These values are then plotted on a set of axes and joined with a smooth curve:

The domain of y= log₁₀x is R+ and the range is R. The line x= 0 is an asymptote.

Check this graph using a graphics calculator.

The graph of y= log₁₀x is a reflection in the line y=x of the graph of y= 10x.

Any pair of functions with this reflection prop-erty are said to be the inverse of each other.

The relationship of log

_a

x

to

a

x

The inverse of a function can be determined by interchanging x and y.

To find the inverse of y=ax:

1. Interchange x and y. x=ay

2. Take log_a of both sides. log_ax= log_aay

3. Use the ‘logarithm of a power’ law to bring the power y to the front of the term. log_ax=ylog_aa

4. But log_aa= 1 so, log_ax=y

Therefore, y= log_ax is the inverse of y=ax.

x 0.1 0.5 1 2 5 10 20

y= log10x -1 -0.30 0 0.30 0.70 1 1.30

y

x

0 1.0 1.5

0.5 2.0

–0.5 –1.0

y = log10x

2 4 6 8 10 12 14 16 18 20

y

x

0 1 1

y = log10x

y = x y = 10x

Find the inverse of y= 5 log₁₀x.

THINK WRITE

Write the rule for the original function. y = 5 log₁₀x Interchange x and y in this rule. x = 5 log₁₀y

Divide both sides by 5. = log₁₀y

Express in index notation. y =

1 2

3 x

5

---4 10

x

5

---1

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The graph of

y

=

log

₂

x

Another common base of logarithms is 2. It is often used in computer science to analyse the complexity of algorithms since computers do all calculations in binary arithmetic (base 2).

Translations of logarithmic graphs

The translation of logarithmic functions is similar to the translation of exponential and other functions.

1. The graph of f(x) = log_a(x+b) is obtained by translating the graph of f(x) = log_a x

horizontally:

b units to the left if b> 0 b units to the right if b< 0.

Note: The asymptote is also translated and its equation is x= −b.

2. The graph of f(x) = log_ax+c is obtained by translating the graph of f(x) = log_ax

vertically:

c units up if c> 0 c units down if c< 0.

3. The graph of f(x) = log_a(x + b) + c is obtained by translating the graph of

f(x) = log_ax b units horizontally and c units vertically as described above.

Sketch the graph of y= 3 log₂(−x) by first setting up a table of values. State the domain and range.

THINK WRITE

Set up a table of values using x = −8, −4, −2, −1, −0.5 as only logarithms of positive values exist, noting the negative sign in front of the x. These values represent powers of 2, namely:

8 = 23, . . . 1 = 20, 0.5 = 2−1. Evaluate y using the given rule

y= 3 log₂(−x). Again, note the negative sign.

Plot the set of points on a set of axes and join them with a smooth curve.

Check the graph using a graphics calculator.

From the graph the domain is R− and the range is R.

Domain is R−. Range is R.

1

x –8 –4 –2 –1 –0.5

y= 3 log₂(–x) 9 6 3 0 –3 2

3 y

x

0 3 6 9

–3

y = 3log2(–x)

–8 –7 –6 –5 –4 –3 –2 –1

4

5

2

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The graph of y= log_a(x− 2) + 3 would have the same basic shape: the value of a

(provided it is greater than 1), controls only the steepness of the graph.

Reflections

The graph of y= −log_ax is a reflection through the x-axis of the graph of y= log_ax.

The graph of y= log_a(−x) is a reflection through the y-axis of the graph of y= log_ax.

Sketch the graph of f(x) = log₁₀(x− 2) + 3 using translation. State the equation of the asymptote.

THINK WRITE

Sketch the basic graph of f(x) = log₁₀x

on a set of axes.

Translate a few points on the graph of

f(x) = log₁₀x, 2 units right and 3 units up.

Translate the graph of f(x) = log₁₀x, 2 units right and 3 units up.

Join the points with a smooth curve in the same shape as f(x) = log₁₀x.

The equation of the asymptote is x= 2. Asymptote is x= 2.

1 y

x

0 1

1

f(x) = log₁₀x

2

y

x

0 1

–1

–2 2 3 4

1 2 3 4 5 6 7

3

2

3 2

2

f(x) = log₁₀x

3

x y

0 1

–1

–2 2 3

1 2 3 4 5 6 7

3

2

3 2

2

f(x) = log10x

f(x) = log10(x – 2) + 3

4

3 WORKED

E

xample

y

x

0 ₁

y = logax, a > 1

y = –logax, a > 1

y

x

0 ₁

– 1

y = log_ax, a > 1

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Extension — Logarithmic graphs

1 Find the inverse of each of the following.

2 Sketch the graphs of each of the following by first completing a table of values. State the domain and range of each.

3 Sketch the graph of each of the following using translation. State the equation of the asymptote in each case.

4

The rule for the graph at right is:

A y= log₁₀(1 −x)

B y= log₁₀ (x− 1)

C y= −log₁₀(1 −x)

D y= −log₁₀(x− 1)

E y= −log₁₀x+ 1

5

The graph of y= log₅2x could be:

A B C D E

a y= 102x b y= 3 log₁₀x c y= 85x

d y= 2 log₃(2x) e y= 100.2x f y= 4 log₁₀(x+ 1)

a y= log₂3x b y= log₂ c y= 2 log₂x

d y= 3 log₁₀x e y= −log₁₀x f y= log₁₀(−x)

a f(x) = log₂(x+ 4) b f(x) = 3 + log₂x c f(x) = log₂(x− 1) + 2

d f(x) = log₂(x− 3) − 2 e f(x) = log₂(2 −x) f f(x) = −log₂(1 −x)

remember

1. f(x) = log_ax is the inverse of g(x) =ax and they are therefore reflections of each other through the line y=x.

2. If a> 1, f(x) = log_ax has: • x-intercept (1, 0) • asymptote x = 0 • domain =R+

• range =R.

3. The graph of f(x) = log_a(x + b) + c is obtained by translating the graph of

f(x) = log_ax b units horizontally and c units vertically.

remember

7.1

W WORKEDORKED

E Examplexample

1

W WORKEDORKED

E Examplexample

2 x

2

---W WORKEDORKED

E Examplexample

3

m

multiple choiceultiple choice

y

x

0 1

m

y

x

0 ₁

y

x

0 ₁

y

x

0 1_– 2

y

x

0 1

y

x

(5)

6

The graph of f(x) = −log₃(x+ 4) could be:

A B C D E

7

The rule for the graph at right is:

A y= log₃(x− 2) + 1

B y= log₃(x+ 2) + 1

C y= log₂(x− 2)

D y= log₃(x− 2)

E y= log₂(x− 2) − 1

m

y

x

0 4

y

x

0 –4

y

x

0 4

y

x

0 –4

y

x

0 –4

m

y

x

0 2 3

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answers

CHAPTER 7 Exponential

functions and logarithms

Exercise 7.1 — Logarithmic graphs

1 2 3 4 C 5 E 6 D 7 A

a y= log₁₀x b y=

c y= log₈x d y=

e y= 5 logx f y=

a

dom =R+, ran =R b

dom =R+, ran =R c

dom =R+, ran =R d

dom =R+, ran =R e

dom =R+, ran =R f

dom =R−, ran =R a

dom = (−4, ∞), ran =R,

x= −4

b

dom = R+, ran =R,

x= 0

c

dom = (1, ∞), ran =R,

x= 1

d

dom = (3, ∞), ran =R,

x= 3

e

dom = (−∞, 2), ran =R,

x= 2

f

dom = (−∞, 1), ran =R,

x= 1

1 2 --- 10 x 3 ---1 5 --- 3 x 2 ---2

----10x4---_–₁

y = log₂3x y

x

0 1_– 3

y

x

0

y = log₂( )x_– 2

2

y

x

0

y = 2 log₂x

1

y

x

0

y = 3 log₁₀2x

1 – 2 y x 0

y = –log₁₀x

1

y x

0

y = log₁₀(–x) –1

y

x

0

y = log

2(x + 4)

–3 –4

y

x

0

y = 3 + log₂x

1 3

y

x

0

y = log

2(x – 1) + 2

1 2

2

y

x

0

y = log₂(x – 3) – 2

1 3 4 2 y x 0

y = log₂(2 – x)

1 2

1

y x

0

y = –log₂(1 – x)

–1 1