4.5: Logarithmic Properties
Using the Product Rule for Logarithms
Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.
\[ \begin{align*} \log_b1 &=0 \\[4pt] \log_bb &=1 \end{align*}\]
For example, \({\log}_51=0\) since \(5^0=1\). And \({\log}_55=1\) since \(5^1=5\).
Next, we have the inverse property.
\[ \begin{align*} \log_b(b^x) &=x \\[4pt] b^{\log_b x} &=x, \,x>0 \end{align*}\]
For example, to evaluate \({\log(100)}\), we can rewrite the logarithm as \({\log}_{10}({10}^2)\), and then apply the inverse property \({\log}_b(b^x)=x\) to get \({\log}_{10}({10}^2)=2\).
To evaluate \(e^{\ln(7)}\), we can rewrite the logarithm as \(e^{{\log}_e7}\), and then apply the inverse property \(b^{{\log}_bx}=x\) to get \(e^{{\log}_e 7}=7\).
Finally, we have the one-to-one property.
\[ \log_bM = \log_bN \text{ if and only if } M=N\]
We can use the one-to-one property to solve the equation \({\log}_3(3x)={\log}_3(2x+5)\) for \(x\). Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for \(x\):
\(3x=2x+5\) Set the arguments equal.
\(x=5\) Subtract \(2x\).
But what about the equation \({\log}_3(3x)+{\log}_3(2x+5)=2\)? The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.
Recall that we use the product rule of exponents to combine the product of exponents by adding: \(x^ax^b=x^{a+b}\). We have a similar property for logarithms, called the product rule for logarithms , which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.
Given any real number \(x\) and positive real numbers \(M\), \(N\), and \(b\), where \(b≠1\), we will show
\({\log}_b(MN)={\log}_b(M)+{\log}_b(N)\).
Let \(m={\log}_bM\) and \(n={\log}_bN\). In exponential form, these equations are \(b^m=M\) and \(b^n=N\). It follows that
\[\begin{align*} {\log}_b(MN)&= {\log}_b(b^mb^n) \qquad \text{Substitute for M and N}\\[4pt] &= {\log}_b(b^{m+n}) \qquad \text{Apply the product rule for exponents}\\[4pt] &= m+n \qquad \text{Apply the inverse property of logs}\\[4pt] &= {\log}_b(M)+{\log}_b(N) \qquad \text{Substitute for m and n} \end{align*}\]
Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider \({\log}_b(wxyz)\). Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:
\({\log}_b(wxyz)={\log}_bw+{\log}_bx+{\log}_by+{\log}_bz\)
The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.
\[\begin{align} {\log}_b(MN)={\log}_b(M)+{\log}_b(N)\text{ for } b> 0 \end{align}\]
- Factor the argument completely, expressing each whole number factor as a product of primes.
- Write the equivalent expression by summing the logarithms of each factor.
Using the Quotient Rule for Logarithms
For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to combine the quotient of exponents by subtracting: \(x^{\frac{a}{b}}=x^{a−b}\). The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms.
The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.
\[{\log}_b\left(\dfrac{M}{N}\right)={\log}_bM−{\log}_bN\]
Just as with the product rule, we can use the inverse property to derive the quotient rule.
Given any real number \(x\) and positive real numbers \(M\), \(N\), and b, b, where \(b≠1\), we will show
\({\log}_b\left(\dfrac{M}{N}\right)={\log}_b(M)−{\log}_b(N)\).
Let \(m={\log}_bM\) and \(n={\log}_bN\). In exponential form, these equations are \(b^m=M\) and \(b^n=N\). It follows that
\[\begin{align*} {\log}_b\left (\dfrac{M}{N} \right )&= {\log}_b\left(\dfrac{b^m}{b^n}\right) \qquad \text{Substitute for M and N}\\[4pt] &= {\log}_b(b^{m-n}) \qquad \text{Apply the quotient rule for exponents}\\[4pt] &= m-n \qquad \text{Apply the inverse property of logs}\\[4pt] &= {\log}_b(M)-{\log}_b(N) \qquad \text{Substitute for m and n} \end{align*}\]
For example, to expand \({\log}\left (\dfrac{2x^2+6x}{3x+9} \right )\), we must first express the quotient in lowest terms. Factoring and canceling we get,
\[\begin{align*} {\log}\left (\dfrac{2x^2+6x}{3x+9} \right )&= {\log}\left (\dfrac{2x(x+3)}{3(x+3)} \right ) \qquad \text{Factor the numerator and denominator}\\[4pt] &= {\log}\left (\dfrac{2x}{3} \right ) \qquad \text{Cancel the common factors} \end{align*}\]
Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule.
\[ \begin{align*} {\log}\left(\dfrac{2x}{3}\right) &={\log}(2x)−{\log}(3) \\[4pt] &={\log}(2)+{\log}(x)−{\log}(3) \end{align*}\]
- Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.
- Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.
- Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.
Using the Power Rule for Logarithms
We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as \(x^2\)? One method is as follows:
\[\begin{align*} {\log}_b(x^2)&= {\log}_b(x\cdot x)\\[4pt] &= {\log}_bx+{\log}_bx\\[4pt] &= 2{\log}_bx \end{align*}\]
Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms , which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,
\(100={10}^2\) \(\sqrt{3}=3^{\dfrac{1}{2}}\) \(\dfrac{1}{e}=e^{−1}\)
The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.
\[{\log}_b(M^n)=n{\log}_bM\]
- Express the argument as a power, if needed.
- Write the equivalent expression by multiplying the exponent times the logarithm of the base.
Expanding Logarithmic Expressions
Taken together, the product rule, quotient rule, and power rule are often called “laws of logs.” Sometimes we apply more than one rule in order to simplify an expression. For example:
\[\begin{align*} {\log}_b \left (\dfrac{6x}{y} \right )&= {\log}_b(6x)-{\log}_by\\[4pt] &= {\log}_b6+{\log}_bx-{\log}_by \end{align*}\]
We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:
\[\begin{align*} {\log}_b\left (\dfrac{A}{C} \right )&= {\log}_b(AC^{-1})\\[4pt] &= {\log}_b(A)+{\log}_b(C^{-1})\\[4pt] &= {\log}_bA+(-1){\log}_bC\\[4pt] &= {\log}_bA−{\log}_bC \end{align*}\]
We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.
With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember, however, that we can only do this with products, quotients, powers, and roots—never with addition or subtraction inside the argument of the logarithm.
Rewrite \(ln \left (\dfrac{x^4y}{7} \right )\) as a sum or difference of logs.
Solution
First, because we have a quotient of two expressions, we can use the quotient rule:
\(\ln \left (\dfrac{x^4y}{7} \right )=\ln(x^4y)−\ln(7)\)
Then seeing the product in the first term, we use the product rule:
\(\ln(x^4y)−\ln(7)=\ln(x^4)+\ln(y)−\ln(7)\)
Finally, we use the power rule on the first term:
\(\ln(x^4)+\ln(y)−\ln(7)=4\ln(x)+\ln(y)−\ln(7)\)
Condensing Logarithmic Expressions
We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will learn later how to change the base of any logarithm before condensing.
- Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.
- Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.
- Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.
Using the Change-of-Base Formula for Logarithms
Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than \(10\) ore, e,we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.
To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms .
Given any positive real numbers \(M\), \(b\), and \(n\), where \(n≠1\) and \(b≠1\),we show
\({\log}_bM=\dfrac{{\log}_nM}{{\log}_nb}\)
Let \(y={\log}_bM\). By taking the log base \(n\) of both sides of the equation, we arrive at an exponential form, namely \(b^y=M\). It follows that
\[\begin{align*} {\log}_n(b^y)&= {\log}_nM \qquad \text{Apply the one-to-one property}\\[4pt] y{\log}_nb&= {\log}_nM \qquad \text{Apply the power rule for logarithms}\\[4pt] y&= \dfrac{{\log}_nM}{{\log}_nb} \qquad \text{Isolate y}\\[4pt] {\log}_bM&= \dfrac{{\log}_nM}{{\log}_nb} \qquad \text{Substitute for y} \end{align*}\]
For example, to evaluate \({\log}_536\) using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.
\[\begin{align*} {\log}_536&= \dfrac{\log(36)}{\log(5)} \qquad \text{Apply the change of base formula using base 10}\\[4pt] &\approx 2.2266 \qquad \text{Use a calculator to evaluate to 4 decimal places} \end{align*}\]
The change-of-base formula can be used to evaluate a logarithm with any base.
For any positive real numbers \(M\), \(b\), and \(n\), where \(n≠1\) and \(b≠1\),
\[{\log}_bM=\dfrac{{\log}_nM}{{\log}_nb}\]
It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.
\[{\log}_bM=\dfrac{\ln M}{\ln b}\]
and
\[{\log}_bM=\dfrac{\log M}{\log b}\]
- Determine the new base \(n\), remembering that the common log, \(\log(x)\), has base 10, and the natural log, \(\ln(x)\),has base \(e\).
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Rewrite the log as a quotient using the change-of-base formula
- The numerator of the quotient will be a logarithm with base \(n\) and argument \(M\).
- The denominator of the quotient will be a logarithm with base \(n\) and argument \(b\).