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# Justifying the logarithm properties

Study the proofs of the logarithm properties: the product rule, the quotient rule, and the power rule.
In this lesson, we will prove three logarithm properties: the product rule, the quotient rule, and the power rule. Before we begin, let's recall a useful fact that will help us along the way.
log, start base, b, end base, left parenthesis, b, start superscript, c, end superscript, right parenthesis, equals, c
In other words, a logarithm in base b reverses the effect of a base b power!
Keep this in mind as you read through the proofs that follow.

## Product Rule: $\log_b(MN)=\log_b(M)+\log_b(N)$log, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis

Let's start by proving a specific case of the rule — the case when M, equals, 4, N, equals, 8, and b, equals, 2.
Substituting these values into log, start base, b, end base, left parenthesis, M, N, right parenthesis, we see:
\begin{aligned}\log_2({4\cdot 8})&=\log_2(2^2\cdot 2^3)&&\small{\gray{2^2=4\text{ and } 2^3=8}}\\ \\ &=\log_2(2^{2+3})&&\small{\gray{\text{a^m\cdot a^n=a^{m+n}}}}\\ \\ &=2+3&&\small{\gray{\text{\log_b(b^c)=c}}}\\ \\ &=\log_2(4)+\log_2(8)&&\small{\gray{\text{Since 2=\log_2(4) and 3=\log_2(8)}}}\\ \end{aligned}
And so we have that log, start base, 2, end base, left parenthesis, 4, dot, 8, right parenthesis, equals, log, start base, 2, end base, left parenthesis, 4, right parenthesis, plus, log, start base, 2, end base, left parenthesis, 8, right parenthesis.
While this only verifies one case, we can follow this logic to prove the product rule in general.
Notice, that writing 4 and 8 as powers of 2 was key to the proof. So in general, we'd like M and N to be powers of the base b. To do this, we can let M, equals, b, start superscript, x, end superscript and N, equals, b, start superscript, y, end superscript for some real numbers x and y.
Then by definition, it is also true that log, start base, b, end base, left parenthesis, M, right parenthesis, equals, x and log, start base, b, end base, left parenthesis, N, right parenthesis, equals, y.
Now we have:
\begin{aligned}\log_b(MN)&=\log_b(b^x\cdot b^y)&&\small{\gray{\text{Substitution}}}\\ \\ &=\log_b(b^{x+y})&&\small{\gray{\text{Properties of exponents}}}\\ \\ &=x+y&&\small{\gray{\text{\log_b(b^c)=c}}} \\\\ &=\log_b(M)+\log_b(N)&&\small{\gray{\text{Substitution}}} \end{aligned}

## Quotient Rule: $\log_b\left(\dfrac{M}{N}\right)=\log_b(M)-\log_b(N)$log, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis

The proof of this property follows a method similar to the one used above.
Again, if we let M, equals, b, start superscript, x, end superscript and N, equals, b, start superscript, y, end superscript, then it follows that log, start base, b, end base, left parenthesis, M, right parenthesis, equals, x and log, start base, b, end base, left parenthesis, N, right parenthesis, equals, y.
We can now prove the quotient rule as follows:
\begin{aligned}\log_b\left(\dfrac{M}{N}\right)&=\log_b\left(\dfrac{b^x}{ b^y}\right)&&\small{\gray{\text{Substitution}}}\\ \\ &=\log_b(b^{x-y})&&\small{\gray{\text{Properties of exponents}}}\\ \\ &=x-y&&\small{\gray{\text{\log_b(b^c)=c}}}\\ \\ &=\log_b(M)-\log_b(N)&&\small{\gray{\text{Substitution}}} \end{aligned}

## Power Rule: $\log_b(M^p)=p\log_b(M)$log, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, log, start base, b, end base, left parenthesis, M, right parenthesis

This time, only M is involved in the property and so it is sufficient to let M, equals, b, start superscript, x, end superscript, which gives us that log, start base, b, end base, left parenthesis, M, right parenthesis, equals, x.
The proof of the power rule is shown below.
\begin{aligned}\log_b\left(M^p\right)&=\log_b(\left({b^x}\right)^p)&&\small{\gray{\text{Substitution}}}\\ \\ &=\log_b(b^{xp})&&\small{\gray{\text{Properties of exponents}}}\\ \\ &=xp&&\small{\gray{\text{\log_b(b^c)=c}}}\\ \\ &=\log_b(M)\cdot p&&\small{\gray{\text{Substitution}}}\\ \\ &=p\cdot \log_b(M)&&\small{\gray{\text{Multiplication is commutative}}} \end{aligned}
Alternatively, we can justify this property by using the product rule.
For example, we know that log, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, dot, M, dot, point, point, point, dot, M, right parenthesis, where M is multiplied by itself p times.
We can now use the product rule along with the definition of multiplication as repeated addition to complete the proof. This is shown below.
\begin{aligned} \log_b(M^p) &=\log_b(M\cdot M\cdot ...\cdot M)&&\small{\gray{\text{Definition of exponents}}}\\ \\ &= \log_b(M)+\log_b(M)+...+\log_b(M)&& \small{\gray{\text{Product rule}}}\\\\ &= p\cdot \log_b(M) &&\small{\gray{\text{Repeated addition is multiplication}}}\end{aligned}
And so you have it! We have just proven the three logarithm properties!