Beyond the Classroom
Properties of Logarithms
There are three important logarithm properties that will allow us to solve
various types of logarithm problems.
1. \(log_a b+log_a c=log_abc\)
2. \(log_ab-log_ac=log_a \frac{b}{c}\)
3. \(log_ab^c=clog_a b\)
From the
History section, we know the first
property led to the inception of logarithms and was a major tool used to
progress society. But why can we adopt these properties as truth? Let's
look at a proof of each of these "rules" to show why these properties
always apply (Khan Academy,
Log rules: Justifying the Logarithm Properties).
The proofs that follow require an understanding of the
connection between logarithms and exponents.
1. \(log_abc=log_ab+log_ac\)
Let's begin by rewriting \(b\) and \(c\) in terms of \(a\). Define
\(b=a^x\) and \(c=a^y\). This definition is valid when we consider that the
range of all exponential function is \((0, \infty)\); therefore, we know
there is a number we can raise \(a\) by to obtain both \(b\) and \(c\) as
long as \(b\) and \(c\) are positive real numbers. Luckily, we know \(b\)
and \(c\) must be positive real numbers because the domain of logarithmic
functions is \((0, \infty)\). Let's substitute these definitions into the
expression.
\(log_abc=log_a (a^x a^y)\)
Employing the properties of exponents, we can rewrite the expression.
\(log_a(a^x a^y )=log_aa^{x+y}\)
Recall the definition of a logarithm established in
Teaching.
"\(log_a b\) is read as 'log base \(a\) of \(b\)' and is the number we must
raise \(a\) by to obtain \(b\)." Using this definition, we can reduce this
expression because the number we raise \(a\) by to obtain \(a^{x+y}\) is
\(x+y\).
\(log_aa^{x+y} =x+y\)
Now, reconsider the definitions \(b=a^x\) and \(c=a^y\). Using our knowledge
of logarithms and exponents, we can rewrite these expressions.
\(b=a^x \Rightarrow x=log_ab\)
\(c=a^y \Rightarrow y=log_ac\)
Making this substitution yields the result we hoped to prove.
\(x+y=log_ab+log_ac\)
A similar proof can be used to show the second property is true.
Note the following string of deductions.
2. \(log_ab-log_ac=log_a\frac{b}{c}\)
Let \(b=a^x\) and \(c=a^y\)
\(log_a \frac{b}{c}=log_a \frac{a^x}{a^y}=log_a a^{x-y}=x-y\)
\(b=a^x \Rightarrow x=log_ab\) and \(c=a^y \Rightarrow y=log_ac\)
\(x-y=log_ab-logac\)
This logarithmic property can also be proven using the connection between
exponents and logarithms.
3. \(log_ab^c=clog_a b\)
Let \(b=a^x\).
\(log_a (a^{x})^{c} = log_a a^{xc}=xc\)
\(b=a^x \Rightarrow x=log_ab\)
\(xc=clog_ab\)
Logarithms as Repeated Division
Another interesting way to look at these logarithmic properties is through
repeated division. This perspective on logarithms has several
advantages, but one of the main one for educators is that they provide a
look at logarithms without having to have a deep conceptual understanding of
exponents. Logarithms as repeated division allow teachers to introduce the idea of
logarithms at younger ages before students learn exponents, or integer
logarithms provide a different way to conceptualize logarithms for students
who are struggling to grasp logarithms as their own entity (MASL, 2019).
Additionally, using something similar to the applet below provides a visual
representation of logarithms.
Before we investigate the properties mentioned above, let's look at an
applet to understand how we can conceptualize logarithms as
repeated division. The questions that follow the applet will help us
understand what this applet is teaching us.
How many times do you have to divide 16 by 2 to obtain 1?
How does this relate to the number of divisions you see below \(log_2 16\)
when you press play (bottom left corner)?
What is the numeric value of \(log_2 16\)?
What do you notice about the relationship between repeated division and
logarithms?
Does this relationship hold true for other values of a? (Move the slider to
find out)
From this applet we can see that logarithms can be seen as the number of
times we must divide the argument by the base to obtain 1.
This look at logarithms is sometimes called integer logarithms because it
falls apart
when the argument of the logarithm is anything other than an integer.
However, logarithms from this perspective are still powerful. To see how,
let's go back to the properties above.
1. \(log_a bc=log_a b+log_a c\)
Consider the example of \(log_2 4+log_2 2\). Using the applet, how many
divisions are below each of these numbers? \(log_2 4=2\). \(log_2 2=1\).
Thus, \(log_2 4+log_2 2=2+1=3\). What number has three divisions below it? Or
in other words, what number must we divide by 2 three times to obtain 1. We
conclude \(log_2 8=3\). Now, look at the arguments. \(2*4=8\). We can see
that this logarithmic property is true from the perspective of integer
logarithms. \(log_2 4 +log_2 2=log_2 (4*2)=log_2 8\).
A similar example shows that the second property also holds true.
2. \(log_a b-log_a c=log_a \frac{b}{c}\)
Consider the example \(log_3 27-log_3 3\). How many times must we divide
each of these arguments by 3 to obtain 1? \(log_3 27=3\). \(log_3 3=1\).
Thus, \(3-1=2\). How could we write the number that can be divided by 3
two times to obtain 1? \(log_3 9=2\). What is \(\frac{27}{3}\)? 9. Thus,
\(log_3 27-log_3 3=log_3 \frac{27}{3}=log_3 9\).
The third property can also be viewed from an integer logarithm perspective.
Note that this property requires a knowledge of exponents. However, it is
worthwhile to evaluate this property from this perspective because the
applet provides a visual way to understand this property. For visual
learners, this may help them understand why this property is really a
property of logarithms.
3. \(log_a b^c=clog_a b\)
Consider the example \(log_2 8\). Using the applet, we can see that there
are three divisions below \(log_2 8\). What is the logarithmic expression
for one of those divisions? \(log_2 2\). So, \(log_2 8\) is 3 of
\(log_2 2\). Or as an equation, \(log_2 8=3log_2 2\). How can we rewrite 8
as an exponential expression with 2 as the base? \(8=2^3\). Thus,
\(log_2 8=log_2 2^3=3log_2 2\).