IB SL MATHS SUMMARY NOTES
LOGS AND EXPONENTIALS
INDEX NOTATION
34
Exponent/ power/ index
Base
INDEX LAWS
FRACTIONAL INDICES
1
𝑛
𝑎 𝑛 = √𝑎
𝑚 1 𝑚 𝑚
𝑛
𝑎𝑛 = (𝑎 )
𝑛 = ( √𝑎 )
EXPANSION AND FACTORISATION
- a(b+c) = ab + ac
- (a+b)(c+d) = ac+ad+bc+bd
- (a+b)(a-b) = a2-b2
- (a+b)2 = a2+2ab+b
- (a-b)2 = a2-2ab+b2
,EXPONENTIAL GRAPHS
𝑓(𝑥) = 𝑎 𝑥 EXPONENTIAL GROWTH FUNCTION 𝑓(𝑥) = 𝑎− 𝑥 EXPONENTIAL DECAY FUNCTION
1 1
Graph passes through (−1, ) and (1, 𝑎) Graph passes through (1, ) and (−1, 𝑎)
𝑎 𝑎
Graph passes through (0,1)
• Domain is all real x
• Range is all real positive numbers
• Y intercept is 1
• No x intercepts
BASE e
Base e = irrational number (2.718 …)
LOGARITHMS
𝑏 = 𝑎 𝑥 then loga b = x
e.g. 8 = 23 then log28 = 3
PROPERTIES OF LOGARITHMS
Rule i.e.
Logaa = 1 𝑎1 = 𝑎
Loga1 = 0 𝑎0 = 1
Loga𝑎𝑛 = n 𝑎𝑛 = 𝑎𝑛
WHEN ARE LOGS UNDEFINED?
Logab if b is negative
Loga0
,LOGARITHMIC FUNCTIONS
𝑦 = 𝑎𝑥
𝑦=𝑥
𝑦 = 𝑙𝑜𝑔a𝑥
Properties of 𝑦 = 𝑙𝑜𝑔a𝑥
• Domain: all positive real numbers
• Range: all real numbers
• X intercept 1
LOG BASE 10 AND NATURAL LOGS
Log to the base 10 Log10x = logx
Log to the base e (natural logs) Logex = lnx
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
a logarithmic function is the inverse of the exponential function
If 𝑓(𝑥) = ⅇ 𝑥 then 𝑓 −1 (𝑥) = 𝑙𝑜𝑔ex
Loga𝑎 𝑥 = alogax = x
Lnex = elnx = x
Log of an exponential/ exponential of a log = x
LOG LAWS
**must learn by <3
Log x + log y = logxy
𝑥
Log x – log y = log
𝑦
Logxn = nlogx
, CHANGE OF BASE
SOLVING EXPONENTIAL EQUATIONS
- Take the ‘log’ of both sides
SOLVING LOGARITHMIC EQUATIONS
1. Ensure that both side of the equation are logarithms with the same base and the equating
the argument (the expression inside the bracket) i.e. “cancelling” the logs
Or
2. By using an exponent
** For both of these methods you must check that the solution/ solutions are possible.
You cannot have the log of a negative number
Sub your solution back into what was in the brackets of the log and then check the result is
positive
LOGS AND EXPONENTIALS
INDEX NOTATION
34
Exponent/ power/ index
Base
INDEX LAWS
FRACTIONAL INDICES
1
𝑛
𝑎 𝑛 = √𝑎
𝑚 1 𝑚 𝑚
𝑛
𝑎𝑛 = (𝑎 )
𝑛 = ( √𝑎 )
EXPANSION AND FACTORISATION
- a(b+c) = ab + ac
- (a+b)(c+d) = ac+ad+bc+bd
- (a+b)(a-b) = a2-b2
- (a+b)2 = a2+2ab+b
- (a-b)2 = a2-2ab+b2
,EXPONENTIAL GRAPHS
𝑓(𝑥) = 𝑎 𝑥 EXPONENTIAL GROWTH FUNCTION 𝑓(𝑥) = 𝑎− 𝑥 EXPONENTIAL DECAY FUNCTION
1 1
Graph passes through (−1, ) and (1, 𝑎) Graph passes through (1, ) and (−1, 𝑎)
𝑎 𝑎
Graph passes through (0,1)
• Domain is all real x
• Range is all real positive numbers
• Y intercept is 1
• No x intercepts
BASE e
Base e = irrational number (2.718 …)
LOGARITHMS
𝑏 = 𝑎 𝑥 then loga b = x
e.g. 8 = 23 then log28 = 3
PROPERTIES OF LOGARITHMS
Rule i.e.
Logaa = 1 𝑎1 = 𝑎
Loga1 = 0 𝑎0 = 1
Loga𝑎𝑛 = n 𝑎𝑛 = 𝑎𝑛
WHEN ARE LOGS UNDEFINED?
Logab if b is negative
Loga0
,LOGARITHMIC FUNCTIONS
𝑦 = 𝑎𝑥
𝑦=𝑥
𝑦 = 𝑙𝑜𝑔a𝑥
Properties of 𝑦 = 𝑙𝑜𝑔a𝑥
• Domain: all positive real numbers
• Range: all real numbers
• X intercept 1
LOG BASE 10 AND NATURAL LOGS
Log to the base 10 Log10x = logx
Log to the base e (natural logs) Logex = lnx
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
a logarithmic function is the inverse of the exponential function
If 𝑓(𝑥) = ⅇ 𝑥 then 𝑓 −1 (𝑥) = 𝑙𝑜𝑔ex
Loga𝑎 𝑥 = alogax = x
Lnex = elnx = x
Log of an exponential/ exponential of a log = x
LOG LAWS
**must learn by <3
Log x + log y = logxy
𝑥
Log x – log y = log
𝑦
Logxn = nlogx
, CHANGE OF BASE
SOLVING EXPONENTIAL EQUATIONS
- Take the ‘log’ of both sides
SOLVING LOGARITHMIC EQUATIONS
1. Ensure that both side of the equation are logarithms with the same base and the equating
the argument (the expression inside the bracket) i.e. “cancelling” the logs
Or
2. By using an exponent
** For both of these methods you must check that the solution/ solutions are possible.
You cannot have the log of a negative number
Sub your solution back into what was in the brackets of the log and then check the result is
positive
