# Sketch the graph of y3 log

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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.

a

### 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 loga of both sides. logax= logaay

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

4. But logaa= 1 so, logax=y

Therefore, y= logax 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

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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) = loga(x+b) is obtained by translating the graph of f(x) = loga 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) = logax+c is obtained by translating the graph of f(x) = logax

vertically:

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

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

f(x) = logax 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

(3)

The graph of y= loga(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= −logax is a reflection through the x-axis of the graph of y= logax.

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

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

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

3

2

3 2

2

f(x) = log10x

f(x) = log10(x – 2) + 3

4

### xample

y

x

0 1

y = logax, a > 1

y = –logax, a > 1

y

x

0 1

– 1

y = logax, a > 1

(4)

### 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) = logax 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) = logax has: • x-intercept (1, 0) • asymptote x = 0 • domain =R+

• range =R.

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

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

## 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

multiple choiceultiple choice

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

multiple choiceultiple choice

y

x

0 4

y

x

0 –4

y

x

0 4

y

x

0 –4

y

x

0 –4

m

multiple choiceultiple choice

y

x

0 2 3

(6)

### 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

References

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