**Logarithmic graphs**

**The graph of **

**y**

**y**

### =

** log**

_{10}**x**

**x**

The graph of *y*= log_{10}*x* 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*= log_{10}*x* 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*=

**log**

_{10}

**x****is a reflection in the**

**line**

*=*

**y**

**x****of the graph of**

*=*

**y****10**

**x****.**

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

**The relationship of log**

_{a}**x**

**x**

** to **

**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 log* _{a}* of both sides. log

*= log*

_{a}x

_{a}ay3. Use the ‘logarithm of a power’ law to bring the power *y* to the front of the term.
log* _{a}x*=

*y*log

_{a}a4. But log* _{a}a*= 1 so, log

*=*

_{a}x*y*

Therefore, *y*= log* _{a}x* is the inverse of

*y*=

*ax*.

* x* 0.1 0.5 1 2 5 10 20

* y*=

**log10**

*-1 -0.30 0 0.30 0.70 1 1.30*

**x***y*

*x*

0 1.0 1.5

0.5 2.0

–0.5 –1.0

*y *= log10*x*

2 4 6 8 10 12 14 16 18 20

*y*

*x*

0 1 1

*y *= log10*x*

*y *= *x*
*y *= 10*x*

**Find the inverse of *** y*=

**5 log**

_{10}

**x****.**

**THINK** **WRITE**

Write the rule for the original function. *y* = 5 log_{10}*x*
Interchange *x* and *y* in this rule. *x* = 5 log_{10}*y*

Divide both sides by 5. = log_{10}*y*

Express in index notation. *y* =

**1**
**2**

**3** *x*

5

**---4** 10

*x*

5

**---1**

**---1**

**The graph of **

**y**

**y**

### =

** log**

_{2}**x**

**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}xhorizontally:

*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* _{a}x*+

*c*is obtained by translating the graph of

*f*(

*x*) = log

_{a}xvertically:

*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_{a}x*b* units horizontally and *c* units vertically as described above.

**Sketch the graph of *** y*=

**3 log**−

_{2}(

**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 log_{2}(−*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 log**–

_{2}(

**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**

**2**

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* _{a}x* is a reflection through the

*x*-axis of the graph of

*y*= log

*.*

_{a}xThe graph of *y*= log* _{a}*(−

*x*) is a reflection through the

*y*-axis of the graph of

*y*= log

*.*

_{a}x**Sketch the graph of ****f****(****x****) **=** log _{10}(**

*−*

**x****2)**+

**3 using translation. State the equation of the**

**asymptote.**

**THINK** **WRITE**

Sketch the basic graph of *f*(*x*) = log_{10}*x*

on a set of axes.

Translate a few points on the graph of

*f*(*x*) = log_{10}*x*, 2 units right and 3 units
up.

Translate the graph of *f*(*x*) = log_{10}*x*, 2 units
right and 3 units up.

Join the points with a smooth curve in
the same shape as *f*(*x*) = log_{10}*x*.

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

**1** *y*

*x*

0 1

1

*f*(*x*) = log_{10}*x*

**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*) = log_{10}*x*

**3**

*x*
*y*

0 1

–1

–2 2 3

1 2 3 4 5 6 7

3

3

2

3 2

2

*f*(*x*) = log10*x*

*f*(*x*) = log10(*x* – 2) + 3

**4**

**3**

**3**

### WORKED

**E**

**xample**

*y*

*x*

0 _{1}

*y* = log*ax*, a > 1

*y* = –log*ax*, a > 1

*y*

*x*

0 _{1}

– 1

*y* = log* _{a}x*, a > 1

**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*= log_{10}(1 −*x*)

**B** *y*= log_{10} (*x*− 1)

**C** *y*= −log_{10}(1 −*x*)

**D** *y*= −log_{10}(*x*− 1)

**E** *y*= −log_{10}*x*+ 1

**5**

The graph of *y*= log_{5}2*x* could be:

**A** **B** **C** **D** **E**

**a** *y*= 102*x* **b** *y*= 3 log_{10}*x* **c** *y*= 85*x*

**d** *y*= 2 log_{3}(2*x*) **e** *y*= 100.2*x* **f** *y*= 4 log_{10}(*x*+ 1)

**a** *y*= log_{2}3*x* **b** *y*= log_{2} **c** *y*= 2 log_{2}*x*

**d** *y*= 3 log_{10}*x* **e** *y*= −log_{10}*x* **f** *y*= log_{10}(−*x*)

**a** *f*(*x*) = log_{2}(*x*+ 4) **b** *f*(*x*) = 3 + log_{2}*x* **c** *f*(*x*) = log_{2}(*x*− 1) + 2

**d** *f*(*x*) = log_{2}(*x*− 3) − 2 **e** *f*(*x*) = log_{2}(2 −*x*) **f** *f*(*x*) = −log_{2}(1 −*x*)

**remember**

1. *f*(*x*) = log* _{a}x* 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* _{a}x* 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_{a}x*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**

**multiple choiceultiple choice**

*y*

*x*

0 _{1}

*y*

*x*

0 _{1}

*y*

*x*

0 1_{–}
2

*y*

*x*

0 1

*y*

*x*

**6**

The graph of *f*(*x*) = −log_{3}(*x*+ 4) could be:

**A** **B** **C** **D** **E**

**7**

The rule for the graph at right is:

**A** *y*= log_{3}(*x*− 2) + 1

**B** *y*= log_{3}(*x*+ 2) + 1

**C** *y*= log_{2}(*x*− 2)

**D** *y*= log_{3}(*x*− 2)

**E** *y*= log_{2}(*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

**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*= log_{10}*x* **b** *y*=

**c** *y*= log_{8}*x* **d** *y*=

**e** *y*= 5 log*x* **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

----10*x*4---_{–}_{1}

*y *= log_{2}3*x*
*y*

*x*

0 1_{–}
3

*y*

*x*

0

*y *= log_{2}( )*x*_{–}
2

2

*y*

*x*

0

*y *= 2 log_{2}*x*

1

*y*

*x*

0

*y *= 3 log_{10}2*x*

1
–
2
*y*
*x*
0

*y *= –log_{10}*x*

1

*y*
*x*

0

*y *= log_{10}(–*x*)
–1

*y*

*x*

0

*y *= log

2(*x* + 4)

–3 –4

*y*

*x*

0

*y *= 3 + log_{2}*x*

1 3

*y*

*x*

0

*y *= log

2(*x* – 1) + 2

1 2

2

*y*

*x*

0

*y *= log_{2}(*x* – 3) – 2

1
3 4
2
*y*
*x*
0

*y *= log_{2}(2 – *x*)

1 2

1

*y*
*x*

0

*y *= –log_{2}(1 – *x*)

–1 1