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