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Here are a few recommended readings before getting started with this lesson.
A logarithm is the inverse function of an exponential function. The logarithm of a positive number m is written as logbm and read as the logarithm of m with base b.
logbm=n⇔bn=m
As a consequence of the definition of a logarithm, two properties can be deduced. In these properties, b is positive and not equal to 1.
Property | Reason |
---|---|
logbb=1 | A number raised to the power of 1 is equal to itself. |
logb1=0 | A number raised to the power of 0 is equal to 1. |
Paulina has recently become excited learning about logarithms.
She eagerly went to her math teacher and asked for some introductory exercises to practice evaluating and rewriting logarithmic expressions. Help her get off to a good start!
Evaluate the logarithms.
Rewrite the logarithmic equations as exponential equations and the exponential equations as logarithmic equations.
logba=c ⇔ a=bc |
---|
a=bc Substitute a=blogba |
logba=c Substitute logbbc=c |
With this information in mind, three properties can be stated.
The logarithm of a product can be written as the sum of the individual logarithms of each factor.
logbmn=logbm+logbn
Rewrite mn as m⋅n
m=blogb(m)
am⋅an=am+n
logb(bm)=m
The logarithm of a quotient can be written as the difference between the logarithm of the numerator and the logarithm of the denominator.
logbnm=logbm−logbn
m=blogb(m)
anam=am−n
logb(bm)=m
The logarithm of a power can be written as the product of the exponent and the logarithm of the base.
logbmn=nlogbm
m=blogb(m)
(am)n=am⋅n
logb(bm)=m
Commutative Property of Multiplication
Split into factors
log2(mn)=log2(m)+log2(n)
log23≈1.585, log25≈2.322
Add terms
log2(ba)=log2(a)−log2(b)
log2(1)=0
log25≈2.322
Subtract term
Split into factors
log2(mn)=log2(m)+log2(n)
Write as a power
log2(am)=m⋅log2(a)
log2(2)=1
Identity Property of Multiplication
log23≈1.585
Add terms
log5(ba)=log5(a)−log5(b)
log5(mn)=log5(m)+log5(n)
log5(am)=m⋅log5(a)
Calculate logarithm
log4(am)=m⋅log4(a)
log4(ba)=log4(a)−log4(b)
log4(mn)=log4(m)+log4(n)
Distribute 21
log4(4)=1
Identity Property of Multiplication
m⋅log2(a)=log2(am)
Calculate power
log2(m)+log2(n)=log2(mn)
Multiply
log2(m)−log2(n)=log2(nm)
Calculate quotient
m⋅log3(a)=log3(am)
Calculate power
log3(m)−log3(n)=log3(nm)
log3(m)+log3(n)=log3(mn)
ca⋅b=ca⋅b