VCE Maths Methods - Unit 1 - Logarithmic functions

Logarithmic functions

• Logarithms• Log rules• Solving logarithmic equations• Graph of a logarithmic function

2

VCE Maths Methods - Unit 1 - Logarithmic functions

Logarithms

• The logarithmic function is the inverse of the exponential function.

• The logarithm is the power (x) required to raise a base (a) to a value (y)

y = a x loga y = x

23=8 log2 8=3

10!1=0.1 log10 0.1=!1

3

VCE Maths Methods - Unit 1 - Logarithmic functions

• The rules for logs are based on the rules for exponential functions

Log rules - sums & di!erences

4

22!25

=27

loga m+loga n = loga (mn) loga m!loga n = loga

mn"

#$

%

&'

log2 4=2 log2 32=5

log2 4+log2 32= log2128

log2128=7

25

23 =22

log2

328

!

"#

$

%&= log2 4

4!32=128

log2 32!log2 8= log2 4

2+5=7 5!3=2

VCE Maths Methods - Unit 1 - Logarithmic functions

loga a =1

Log rules

5

log2 2=1

21=2

log21=0

20=1

22( )3=26

log2 26=6log2 22

=6 log2

18= log2 2!3

log2 2!3=!3

loga 1=0

loga an=n loga a

loga1x=!loga x

18=2!3

log2 22( )3=3log2 22

=3!2=6

VCE Maths Methods - Unit 1 - Logarithmic functions

Solving logarithmic equations

log3 81 log2 40 ! log25

=

log3 125log3 5

= log3 34

=4log3 3

=4

= log2

405

= log2 8

= log223

=3log22

=3

=

log3 53

log3 51

=

3log3 51log3 5

=3

6

VCE Maths Methods - Unit 1 - Logarithmic functions

Solving for other bases

• Most calculators only calculate the log of base 10 & base e (Euler’s number = 2.718...).

• Other bases can be calculated from these. (You can use either e or 10)

2x=1024 log21024= x

log10 2x= log10 1024

x log10 2= log10 1024

x = log10 1024

log10 2

x = 3.0103

0.30103

x =10

1.05x=2

log10 1.05x= log10 2

x log10 1.05= log10 2

x = log10 2

log10 1.05

x = 0.30103

0.02120

x =14.2

5% growth - how long does it take to double?

7

VCE Maths Methods - Unit 1 - Logarithmic functions

Graph of a logarithmic function

8

y = log2 x

y =2x

y = x

(1,0)

Asymptote at x = 0

Asymptote at y = 0

(0,1)

22=4log24=2