4.7: Logarithmic Properties

Learning Objectives

• Use the product rule for logarithms.
• Use the quotient rule for logarithms.
• Use the power rule for logarithms.
• Expand logarithmic expressions.
• Condense logarithmic expressions.
• Use the change-of-base formula for logarithms.

In chemistry, the pH scale is used as a measure of the acidity or alkalinity of a substance. Substances with a pH less than \(7\) are considered acidic, and substances with a pH greater than \(7\) are said to be alkaline. Our bodies, for instance, must maintain a pH close to \(7.35\) in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances:

• Battery acid: \(0.8\)
• Stomach acid: \(2.7\)
• Orange juice: \(3.3\)
• Pure water: \(7\) at \(25^\circ C\)
• Human blood: \(7.35\)
• Fresh coconut: \(7.8\)
• Sodium hydroxide (lye): \(14\)

To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where \([\ce{H^{+}}]\) is the concentration of hydrogen ions in the solution

\[\begin{align} {pH}&=−{\log}([\ce{H^{+}}]) \label{eq1} \\[4pt] &={\log}\left(\dfrac{1}{[\ce{H^{+}}]}\right) \label{eq2} \end{align}\]

The equivalence of Equations \ref{eq1} and \ref{eq2} is one of the logarithm properties we will examine in this section.

Figure \(\PageIndex{1}\): The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan).

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.

Subtract \(2x\)

\(x=5\)

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:

\({\log}_b(wxyz)={\log}_bw+{\log}_bx+{\log}_by+{\log}_bz\)

Example \(\PageIndex{1}\): Using the Product Rule for Logarithms

Expand \({\log}_3(30x(3x+4))\).

Solution

We begin by factoring the argument completely, expressing \(30\) as a product of primes.

\({\log}_3(30x(3x+4))={\log}_3(2⋅3⋅5⋅x⋅(3x+4))\)

Next we write the equivalent equation by summing the logarithms of each factor.

\({\log}_3(30x(3x+4))={\log}_3(2)+{\log}_3(3)+{\log}_3(5)+{\log}_3(x)+{\log}_3(3x+4)\)

Exercise \(\PageIndex{1}\)

Expand \({\log}_b(8k)\).

Answer

We begin by factoring the argument completely, expressing \(8\) as a product of primes.

\({\log}_b(8k)={\log}_b(2⋅2⋅2⋅k)\)

Next we write the equivalent equation by summing the logarithms of each factor.

\({\log}_b2+{\log}_b2+{\log}_b2+{\log}_bk=3{\log}_b2+{\log}_bk\)

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

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.

Proof

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.

\[ \begin{align*} {\log}\left(\dfrac{2x}{3}\right) &={\log}(2x)−{\log}(3) \\[4pt] &={\log}(2)+{\log}(x)−{\log}(3) \end{align*}\]

Example \(\PageIndex{2}\): Using the Quotient Rule for Logarithms

Expand \({log}_2\left(\dfrac{15x(x−1)}{(3x+4)(2−x)}\right)\).

Solution

First we note that the quotient is factored and in lowest terms, so we apply the quotient rule.

\({\log}_2(\dfrac{15x(x−1)}{(3x+4)(2−x)})={\log}_2(15x(x−1))−{\log}_2((3x+4)(2−x))\)

Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that the prime factors of the factor 15 are 3 and 5.

\[\begin{align*} {\log}_2(15x(x-1))-{\log}_2((3x+4)(2-x))&= [{\log}_2(3)+{\log}_2(5)+{\log}_2(x)+{\log}_2(x-1)]-[{\log}_2(3x+4)+{\log}_2(2-x)]\\[4pt] &= {\log}_2(3)+{\log}_2(5)+{\log}_2(x)+{\log}_2(x-1)-{\log}_2(3x+4)-{\log}_2(2-x) \end{align*}\]

Analysis

There are exceptions to consider in this and later examples. First, because denominators must never be zero, this expression is not defined for \(x=−\dfrac{4}{3}\) and \(x=2\). Also, since the argument of a logarithm must be positive, we note as we observe the expanded logarithm, that \(x>0\), \(x>1\), \(x>−\dfrac{4}{3}\), and \(x<2\). Combining these conditions is beyond the scope of this section, and we will not consider them here or in subsequent exercises.

Exercise \(\PageIndex{2}\)

Expand \({\log}_3\left(\dfrac{7x^2+21x}{7x(x−1)(x−2)}\right)\).

Answer

First, we note that a \(7x\) can be factored out of the numerator and denominator:  \({\log}_3\left(\dfrac{x+3}{(x−1)(x−2)}\right)\).  

Now we apply the quotient rule of logarithms:  \[\begin{align*} {\log}_3(x+3)-[{\log}_3(x-1)+{\log}_3(x-2)] &=\({\log}_3(x+3)−{\log}_3(x−1)−{\log}_3(x−2)\) \end{align*}\]

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

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.

\[{\log}_b(M^n)=n{\log}_bM\]

Example \(\PageIndex{3}\): Expanding a Logarithm with Powers

Expand \({\log}_2x^5\).

Solution

The argument is already written as a power, so we identify the exponent, 5, and the base, \(x\), and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

\({\log}_2(x^5)=5{\log}_2x\)

Example \(\PageIndex{4}\): Rewriting an Expression as a Power before Using the Power Rule

Expand \({\log}_3(25)\) using the power rule for logs.

Solution

Expressing the argument as a power, we get \({\log}_3(25)={\log}_3(5^2)\).

Next we identify the exponent, \(2\), and the base, \(5\), and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

\({\log}_3(52)=2{\log}_3(5)\)

Exercise \(\PageIndex{3}\)

Expand \(\ln\left (\dfrac{1}{x^2} \right )\).

Answer

\(−2\ln(x)\)

Example \(\PageIndex{5}\): Using the Power Rule in Reverse

Rewrite \(4\ln(x)\) using the power rule for logs to a single logarithm with a leading coefficient of \(1\).

Solution

Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression \(4\ln(x)\), we identify the factor, \(4\), as the exponent and the argument, \(x\), as the base, and rewrite the product as a logarithm of a power: \(4\ln(x)=\ln(x^4)\).

Exercise \(\PageIndex{4}\)

Rewrite \(2{\log}_34\) using the power rule for logs to a single logarithm with a leading coefficient of \(1\).

Answer

\({\log}_316\)

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.

You may be wondering if we can expand \(\ln(x^2+y^2)\) No. There is no way to expand the logarithm of a sum or difference inside the argument of the logarithm.

Example \(\PageIndex{6}\): Expanding Logarithms Using Product, Quotient, and Power Rules

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:

\(\ln(x^4)+\ln(y)−\ln(7)=4\ln(x)+\ln(y)−\ln(7)\)

Exercise \(\PageIndex{5}\)

Expand \(\log \left (\dfrac{x^2y^3}{z^4} \right )\).

Answer

\(2\log x+3\log y−4\log z\)

Example \(\PageIndex{7}\): Expanding Complex Logarithmic Expressions

Expand \({\log}_6 \left (\dfrac{64x^3(4x+1)}{(2x−1)} \right )\).

Solution

We can expand by applying the Product and Quotient Rules.

\[\begin{align*} {\log}_6\left (\dfrac{64x^3(4x+1)}{(2x-1)} \right )&= {\log}_664+{\log}_6x^3+{\log}_6(4x+1)-{\log}_6(2x-1)\qquad \text{Apply the Quotient Rule}\\[4pt] &= {\log}_626+{\log}_6x^3+{\log}_6(4x+1)-{\log}_6(2x-1) \qquad \text{Simplify by writing 64 as } 2^6\\[4pt] &= 6{\log}_62+3{\log}_6x+{\log}_6(4x+1)-{\log}_6(2x-1) \qquad \text{Apply the Power Rule} \end{align*}\]

Exercise \(\PageIndex{6}\)

Expand \(\ln \left (\dfrac{\sqrt{(x−1){(2x+1)}^2}}{(x^2−9)}\right )\).

Answer

\(\dfrac{1}{2}\ln(x−1)+\ln(2x+1)−\ln(x+3)−\ln(x−3)\)

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.

Example \(\PageIndex{8}\): Using the Product and Quotient Rules to Combine Logarithms

Write \({\log}_3(5)+{\log}_3(8)−{\log}_3(2)\) as a single logarithm.

Solution

Using the product and quotient rules

\({\log}_3(5)+{\log}_3(8)={\log}_3(5⋅8)={\log}_3(40)\)

This reduces our original expression to

\({\log}_3(40)−{\log}_3(2)\)

Then, using the quotient rule

\({\log}_3(40)−{\log}_3(2)={\log}_3 \left (\dfrac{40}{2} \right )={\log}_3(20)\)

Exercise \(\PageIndex{7}\)

Condense \({\log}_2(x^2)+\dfrac{1}{2}{\log}_2(x−1)−3{\log}_2({(x+3)}^2)\).

Answer

We apply the power rule first:

\({\log}_2(x^2)+\dfrac{1}{2}{\log}_2(x−1)−3{\log}_2({(x+3)}^2)={\log}_2(x^2)+{\log}_2(\sqrt{x−1})−{\log}_2({(x+3)}^6)\)

Next we apply the product rule to the sum:

\({\log}_2(x^2)+{\log}_2(\sqrt{x−1})−{\log}_2({(x+3)}^6)={\log}_2(x^2\sqrt{x−1})−{\log}_2({(x+3)}^6)\)

Finally, we apply the quotient rule to the difference:

\({\log}_2(x^2\sqrt{x−1})−{\log}_2({(x+3)}^6)={\log}_2\dfrac{x^2\sqrt{x−1}}{{(x+3)}^6}\)

Example \(\PageIndex{9}\): Rewriting as a Single Logarithm

Rewrite \(2\log x−4\log(x+5)+\dfrac{1}{x}\log(3x+5)\) as a single logarithm.

Solution

We apply the power rule first:

\(2\log x−4\log(x+5)+\dfrac{1}{x}\log(3x+5)=\log(x^2)−\log{(x+5)}^4+\log({(3x+5)}^{x^{−1}})\)

Next we rearrange and apply the product rule to the sum:

\[\begin{align*} \log(x^2)-\log{(x+5)}^4+\log({(3x+5)}^{x^{-1}})&= \log(x^2)+\log({(3x+5)}^{x^{-1}}-\log{(x+5)}^4\\[4pt] &= \log(x^2{(3x+5)}^{x^{-1}})-\log{(x+5)}^4\\[4pt] &= \log\dfrac{x^2{(3x+5)}^{x^{-1}}}{{(x+5)}^4} \qquad \text{Apply the quotient rule to the difference} \end{align*}\]

Exercise \(\PageIndex{8}\)

Rewrite \(\log(5)+0.5\log(x)−\log(7x−1)+3\log(x−1)\) as a single logarithm.

Answer

\(\log \dfrac{5{(x−1)}^3\sqrt{x}}{(7x−1)}\

Example \(\PageIndex{10}\): Applying of the Laws of Logs

Recall that, in chemistry, \({pH}=−\log[H+]\). If the concentration of hydrogen ions in a liquid is doubled, what is the effect on pH?

Solution

Suppose \(C\) is the original concentration of hydrogen ions, and \(P\) is the original pH of the liquid. Then \(P=–\log(C)\). If the concentration is doubled, the new concentration is \(2C\). Then the pH of the new liquid is

\(pH=−\log(2C)\)

Using the product rule of logs

\(pH=−\log(2C)=−(\log(2)+\log(C))=−\log(2)−\log(C)\)

Since \(P=–\log(C)\),the new pH is

\(pH=P−\log(2)≈P−0.301\)

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.

\[\begin{align*} {\log}_536&= \dfrac{\log(36)}{\log(5)} \qquad \text{Apply the change of base formula using base 10}\\[4pt] &\approx 2.2266 \qquad \text{Use a calculator to evaluate to 4 decimal places} \end{align*}\]

Example \(\PageIndex{11}\): Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Change \({\log}_53\) to a quotient of natural logarithms.

Solution

Because we will be expressing \({\log}_53\) as a quotient of natural logarithms, the new base, \(n=e\).

We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument \(3\). The denominator of the quotient will be the natural log with argument 5.

\({\log}_bM=\dfrac{\ln M}{\ln b}\)

\({\log}_53=\dfrac{\ln3}{\ln5}\)

Key Concepts

The Product Rule for Logarithms	\({\log}_b(MN)={\log}_b(M)+{\log}_b(N)\)
The Quotient Rule for Logarithms	\({\log}_b(\dfrac{M}{N})={\log}_bM−{\log}_bN\)
The Power Rule for Logarithms	\({\log}_b(M^n)=n{\log}_bM\)
The Change-of-Base Formula	\({\log}_bM=\dfrac{{\log}_nM}{{\log}_nb}\) \(n>0\), \(n≠1\), \(b≠1\)

• We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. 
• We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. 
• We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. 
• We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. 
• The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm.
• We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. 
• The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and \(e\) as the quotient of natural or common logs. That way a calculator can be used to evaluate.