Use the Properties of Logarithms

Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.

The first two properties derive from the definition of logarithms. Since $a^{0} = 1,$ we can convert this to logarithmic form and get ${log}_{a} 1 = 0 .$ Also, since $a^{1} = a,$ we get ${log}_{a} a = 1 .$

In the next example we could evaluate the logarithm by converting to exponential form, as we have done previously, but recognizing and then applying the properties saves time.

Evaluate using the properties of logarithms: ⓐ ${log}_{8} 1$ and ⓑ ${log}_{6} 6 .$

Solution

ⓐ

Table demonstrating the evaluation of log_8 1, using the logarithmic property log_a 1 = 0 to show the result is 0.
	${log}_{8} 1$
Use the property, ${log}_{a} 1 = 0$ .	$0 {log}_{8} 1 = 0$

ⓑ
$\begin{matrix} {log}_{6} 6 \\ Use the property, {log}_{a} a = 1 . & 1 & {log}_{6} 6 = 1 \end{matrix}$

The next two properties can also be verified by converting them from exponential form to logarithmic form, or the reverse.

The exponential equation $a^{{log}_{a} x} = x$ converts to the logarithmic equation ${log}_{a} x = {log}_{a} x,$ which is a true statement for positive values for x only.

The logarithmic equation ${log}_{a} a^{x} = x$ converts to the exponential equation $a^{x} = a^{x},$ which is also a true statement.

These two properties are called inverse properties because, when we have the same base, raising to a power “undoes” the log and taking the log “undoes” raising to a power. These two properties show the composition of functions. Both ended up with the identity function which shows again that the exponential and logarithmic functions are inverse functions.

In the next example, apply the inverse properties of logarithms.

Evaluate using the properties of logarithms: ⓐ $4^{{log}_{4} 9}$ and ⓑ ${log}_{3} 3^{5} .$

Solution

ⓐ

Demonstration of the logarithmic property a^(log_a x) = x using an example.
	$4^{{log}_{4} 9}$
Use the property, $a^{{log}_{a} x} = x$ .	$9 4^{{log}_{4} 9} = 9$

ⓑ

This table illustrates a logarithmic expression and a related property, showing a step in its calculation.
	${log}_{3} 3^{5}$
Use the property, $a^{{log}_{a} x} = x$ .	$5 {log}_{3} 3^{5} = 5$

There are three more properties of logarithms that will be useful in our work. We know exponential functions and logarithmic function are very interrelated. Our definition of logarithm shows us that a logarithm is the exponent of the equivalent exponential. The properties of exponents have related properties for exponents.

In the Product Property of Exponents, $a^{m} \cdot a^{n} = a^{m + n},$ we see that to multiply the same base, we add the exponents. The Product Property of Logarithms, ${log}_{a} M \cdot N = {log}_{a} M + {log}_{a} N$ tells us to take the log of a product, we add the log of the factors.

We use this property to write the log of a product as a sum of the logs of each factor.

Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible: ⓐ ${log}_{3} 7 x$ and ⓑ ${log}_{4} 64 x y .$

Solution

ⓐ

Demonstrates expanding a logarithmic expression log_3(7x) into log_3(7) + log_3(x) using the Product Property of logarithms.
	${log}_{3} 7 x$
Use the Product Property, ${log}_{a} (M \cdot N) = {log}_{a} M + {log}_{a} N$ .	${log}_{3} 7 + {log}_{3} x$
	${log}_{3} 7 x = {log}_{3} 7 + {log}_{3} x$

ⓑ

This table demonstrates the step-by-step expansion of a logarithmic expression, log4(64xy), using the product property of logarithms, simplifying it to 3 + log4(x) + log4(y).
	${log}_{4} 64 x y$
Use the Product Property, ${log}_{a} (M \cdot N) = {log}_{a} M + {log}_{a} N$ .	${log}_{4} 64 + {log}_{4} x + {log}_{4} y$
Simplify by evaluating ${log}_{4} 64$ .	$3 + \log_{4} x + \log_{4} y$
	${log}_{4} 64 x y = 3 + {log}_{4} x + {log}_{4} y$

Similarly, in the Quotient Property of Exponents, $\frac{a^{m}}{a^{n}} = a^{m - n},$ we see that to divide the same base, we subtract the exponents. The Quotient Property of Logarithms, ${log}_{a} \frac{M}{N} = {log}_{a} M - {log}_{a} N$ tells us to take the log of a quotient, we subtract the log of the numerator and denominator.

Note that ${log}_{a} M - {log}_{a} N \neq {log}_{a} (M - N) .$

We use this property to write the log of a quotient as a difference of the logs of each factor.

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.
ⓐ ${log}_{5} \frac{5}{7}$ and ⓑ $log \frac{x}{100}$

Solution

ⓐ

Step-by-step simplification of the logarithmic expression log5(5/7) using the Quotient Property.
	${log}_{5} \frac{5}{7}$
Use the Quotient Property, ${log}_{a} \frac{M}{N} = {log}_{a} M - {log}_{a} N$ .	${log}_{5} 5 - {log}_{5} 7$
Simplify.	$1 - {log}_{5} 7$
	${log}_{5} \frac{5}{7} = 1 - {log}_{5} 7$

ⓑ

Step-by-step expansion of log(x/100) using the logarithm quotient property and simplification.
	$log \frac{x}{100}$
Use the Quotient Property, ${log}_{a} \frac{M}{N} = {log}_{a} M - {log}_{a} N$ .	$log x - log 100$
Simplify.	$log x - 2$
	$log \frac{x}{100} = log x - 2$

The third property of logarithms is related to the Power Property of Exponents, ${(a^{m})}^{n} = a^{m \cdot n},$ we see that to raise a power to a power, we multiply the exponents. The Power Property of Logarithms, ${log}_{a} M^{p} = p {log}_{a} M$ tells us to take the log of a number raised to a power, we multiply the power times the log of the number.

We use this property to write the log of a number raised to a power as the product of the power times the log of the number. We essentially take the exponent and throw it in front of the logarithm.

Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.
ⓐ ${log}_{5} 4^{3}$ and ⓑ $log x^{10}$

Solution

ⓐ

Illustration of the Power Property of logarithms with an example.
	${log}_{5} 4^{3}$
Use the Power Property, ${log}_{a} M^{p} = p {log}_{a} M$ .	$3 {log}_{5} 4$
	${log}_{5} 4^{3} = 3 {log}_{5} 4$

ⓑ

Application of the Power Property of Logarithms to transform log(x^10) into 10log(x).
	$log x^{10}$
Use the Power Property, ${log}_{a} M^{p} = p {log}_{a} M$ .	$10 log x$
	$log x^{10} = 10 log x$

We summarize the Properties of Logarithms here for easy reference. While the natural logarithms are a special case of these properties, it is often helpful to also show the natural logarithm version of each property.

Now that we have the properties we can use them to “expand” a logarithmic expression. This means to write the logarithm as a sum or difference and without any powers.

We generally apply the Product and Quotient Properties before we apply the Power Property.

Use the Properties of Logarithms to expand the logarithm ${log}_{4} (2 x^{3} y^{2})$ . Simplify, if possible.

Solution

Step-by-step expansion of the logarithmic expression log4(2x3y2) using logarithm properties.
	${log}_{4} (2 x^{3} y^{2})$
Use the Product Property, ${log}_{a} M \cdot N = {log}_{a} M + {log}_{a} N$ .	${log}_{4} 2 + {log}_{4} x^{3} + {log}_{4} y^{2}$
Use the Power Property, ${log}_{a} M^{p} = p {log}_{a} M$ , on the last two terms.	${log}_{4} 2 + 3 {log}_{4} x + 2 {log}_{4} y$
Simplify.	$\frac{1}{2} + 3 {log}_{4} x + 2 {log}_{4} y$
	${log}_{4} (2 x^{3} y^{2}) = \frac{1}{2} + 3 {log}_{4} x + 2 {log}_{4} y$

When we have a radical in the logarithmic expression, it is helpful to first write its radicand as a rational exponent.

Use the Properties of Logarithms to expand the logarithm ${log}_{2} \sqrt[4]{\frac{x^{3}}{3 y^{2} z}}$ . Simplify, if possible.

Solution

Step-by-step expansion of a complex logarithmic expression using properties of logarithms.
	${log}_{2} \sqrt[4]{\frac{x^{3}}{3 y^{2} z}}$
Rewrite the radical with a rational exponent.	${log}_{2} {(\frac{x^{3}}{3 y^{2} z})}^{\frac{1}{4}}$
Use the Power Property, ${log}_{a} M^{p} = p {log}_{a} M$ .	$\frac{1}{4} {log}_{2} (\frac{x^{3}}{3 y^{2} z})$
Use the Quotient Property, ${log}_{a} M \cdot N = {log}_{a} M - {log}_{a} N$ .	$\frac{1}{4} ({log}_{2} (x^{3}) - {log}_{2} (3 y^{2} z))$
Use the Product Property, ${log}_{a} M \cdot N = {log}_{a} M + {log}_{a} N$ , in the second term.	$\frac{1}{4} ({log}_{2} (x^{3}) - ({log}_{2} 3 + {log}_{2} y^{2} + {log}_{2} z))$
Use the Power Property, ${log}_{a} M^{p} = p {log}_{a} M$ , inside the parentheses.	$\frac{1}{4} (3 {log}_{2} x - ({log}_{2} 3 + 2 {log}_{2} y + {log}_{2} z))$
Simplify by distributing.	$\frac{1}{4} (3 {log}_{2} x - {log}_{2} 3 - 2 {log}_{2} y - {log}_{2} z)$
	${log}_{2} \sqrt[4]{\frac{x^{3}}{3 y^{2} z}} = \frac{1}{4} (3 {log}_{2} x - {log}_{2} 3 - 2 {log}_{2} y - {log}_{2} z)$

The opposite of expanding a logarithm is to condense a sum or difference of logarithms that have the same base into a single logarithm. We again use the properties of logarithms to help us, but in reverse.

To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.

Use the Properties of Logarithms to condense the logarithm ${log}_{4} 3 + {log}_{4} x - {log}_{4} y$ . Simplify, if possible.

Solution

Simplifying a logarithmic expression using the Product and Quotient Properties of logarithms, step-by-step.
The log expressions all have the same base, 4.	${log}_{4} 3 + {log}_{4} x - {log}_{4} y$
The first two terms are added, so we use the Product Property, ${log}_{a} M + {log}_{a} N = {log}_{a} M \cdot N$ .	${log}_{4} 3 x - {log}_{4} y$
Since the logs are subtracted, we use the Quotient Property, ${log}_{a} M - {log}_{a} N = {log}_{a} \frac{M}{N}$ .	${log}_{4} \frac{3 x}{y}$
	${log}_{4} 3 + {log}_{4} x - {log}_{4} y = {log}_{4} \frac{3 x}{y}$

Use the Properties of Logarithms to condense the logarithm $2 {log}_{3} x + 4 {log}_{3} (x + 1)$ . Simplify, if possible.

Solution

Steps to combine a sum of logarithmic expressions into a single logarithm using the Power and Product Properties.
The log expressions have the same base, 3.	$2 {log}_{3} x + 4 {log}_{3} (x + 1)$
Use the Power Property, ${log}_{a} M + {log}_{a} N = {log}_{a} M \cdot N$ .	${log}_{3} x^{2} + {log}_{3} {(x + 1)}^{4}$
The terms are added, so we use the Product Property, ${log}_{a} M + {log}_{a} N = {log}_{a} M \cdot N$ .	${log}_{3} x^{2} {(x + 1)}^{4}$
	$2 {log}_{3} x + 4 {log}_{3} (x + 1) = {log}_{3} x^{2} {(x + 1)}^{4}$

Use the Change-of-Base Formula

To evaluate a logarithm with any other base, we can use the Change-of-Base Formula. We will show how this is derived.

This table illustrates the step-by-step derivation of the logarithm change of base formula using algebraic manipulation.
Suppose we want to evaluate ${log}_{a} M$ .	${log}_{a} M$
Let $y = {log}_{a} M$ .	$y = {log}_{a} M$
Rewrite the expression in exponential form.	$a^{y} = M$
Take the ${log}_{b}$ of each side.	${log}_{b} a^{y} = {log}_{b} M$
Use the Power Property.	$y {log}_{b} a = {log}_{b} M$
Solve for $y$ .	$y = \frac{{log}_{b} M}{{log}_{b} a}$
Substitute $y = {log}_{a} M$ .	${log}_{a} M = \frac{{log}_{b} M}{{log}_{b} a}$

The Change-of-Base Formula introduces a new base $b .$ This can be any base b we want where $b > 0, b \neq 1 .$ Because our calculators have keys for logarithms base 10 and base e, we will rewrite the Change-of-Base Formula with the new base as 10 or e.

When we use a calculator to find the logarithm value, we usually round to three decimal places. This gives us an approximate value and so we use the approximately equal symbol $(≈)$ .

Rounding to three decimal places, approximate ${log}_{4} 35 .$

Solution


Use the Change-of-Base Formula.	$The image displays the logarithm change of base formula: log subscript 'a' of M equals the fraction of log subscript 'b' of M over log subscript 'b' of 'a'.$
Identify a and M. Choose 10 for b.
Enter the expression $\frac{log 35}{log 4}$ in the calculator using the log button for base 10. Round to three decimal places.

Key Concepts

Properties of Logarithms

${log}_{a} 1 = 0 {log}_{a} a = 1$
Inverse Properties of Logarithms
- For $a > 0,$ $x > 0$ and $a \neq 1$
  
  $a^{{log}_{a} x} = x {log}_{a} a^{x} = x$
Product Property of Logarithms
- If $M > 0, N > 0, a > 0$ and $a \neq 1,$ then,
  
  ${log}_{a} M \cdot N = {log}_{a} M + {log}_{a} N$
  
  The logarithm of a product is the sum of the logarithms.
Quotient Property of Logarithms
- If $M > 0, N > 0, a > 0$ and $a \neq 1,$ then,
  
  ${log}_{a} \frac{M}{N} = {log}_{a} M - {log}_{a} N$
  
  The logarithm of a quotient is the difference of the logarithms.
Power Property of Logarithms
- If $M > 0, a > 0, a \neq 1$ and $p$ is any real number then,
  
  ${log}_{a} M^{p} = p {log}_{a} M$
  
  The log of a number raised to a power is the product of the power times the log of the number.

Properties of Logarithms Summary
If

M > 0, a > 0, a \neq 1

and

p

is any real number then,

Property	Base $a$	Base $e$
	${log}_{a} 1 = 0$	$ln 1 = 0$
	${log}_{a} a = 1$	$ln e = 1$
Inverse Properties	$\begin{matrix} a^{{log}_{a} x} = x \\ {log}_{a} a^{x} = x \end{matrix}$	$\begin{matrix} e^{ln x} = x \\ ln e^{x} = x \end{matrix}$
Product Property of Logarithms	${log}_{a} (M \cdot N) = {log}_{a} M + {log}_{a} N$	$ln (M \cdot N) = ln M + ln N$
Quotient Property of Logarithms	${log}_{a} \frac{M}{N} = {log}_{a} M - {log}_{a} N$	$ln \frac{M}{N} = ln M - ln N$
Power Property of Logarithms	${log}_{a} M^{p} = p {log}_{a} M$	$ln M^{p} = p ln M$

Change-of-Base Formula
For any logarithmic bases a and b, and $M > 0,$

$\begin{matrix} {log}_{a} M = \frac{{log}_{b} M}{{log}_{b} a} & {log}_{a} M = \frac{log M}{log a} & {log}_{a} M = \frac{ln M}{ln a} \\ new base b & new base 10 & new base e \end{matrix}$

Practice Makes Perfect

Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

ⓐ ${log}_{4} 1$ ⓑ ${log}_{8} 8$

ⓐ ${log}_{12} 1$ ⓑ $ln e$

Solution

ⓐ 0 ⓑ 1

ⓐ $3^{{log}_{3} 6}$ ⓑ ${log}_{2} 2^{7}$

ⓐ $5^{{log}_{5} 10}$ ⓑ ${log}_{4} 4^{10}$

Solution

ⓐ 10 ⓑ 10

ⓐ $8^{{log}_{8} 7}$ ⓑ ${log}_{6} 6^{−2}$

ⓐ $6^{{log}_{6} 15}$ ⓑ ${log}_{8} 8^{−4}$

Solution

ⓐ 15 ⓑ $−4$

ⓐ $10^{log \sqrt{5}}$ ⓑ $log 10^{−2}$

ⓐ $10^{log \sqrt{3}}$ ⓑ $log 10^{−1}$

Solution

ⓐ $\sqrt{3}$ ⓑ $−1$

ⓐ $e^{ln 4}$ ⓑ $ln e^{2}$

ⓐ $e^{ln 3}$ ⓑ $ln e^{7}$

Solution

ⓐ 3 ⓑ 7

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${log}_{4} 6 x$

${log}_{5} 8 y$

Solution

${log}_{5} 8 + {log}_{5} y$

${log}_{2} 32 x y$

${log}_{3} 81 x y$

Solution

$4 + {log}_{3} x + {log}_{3} y$

$log 100 x$

$log 1000 y$

Solution

$3 + log y$

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

${log}_{3} \frac{3}{8}$

${log}_{6} \frac{5}{6}$

Solution

${log}_{6} 5 - 1$

${log}_{4} \frac{16}{y}$

${log}_{5} \frac{125}{x}$

Solution

$3 - {log}_{5} x$

$log \frac{x}{10}$

$log \frac{10,000}{y}$

Solution

$4 - log y$

$ln \frac{e^{3}}{3}$

$ln \frac{e^{4}}{16}$

Solution

$4 - ln 16$

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.

${log}_{3} x^{2}$

${log}_{2} x^{5}$

Solution

$5 {log}_{2} x$

$log x^{−2}$

$log x^{−3}$

Solution

$−3 log x$

${log}_{4} \sqrt{x}$

${log}_{5} \sqrt[3]{x}$

Solution

$\frac{1}{3} {log}_{5} x$

$ln x^{\sqrt{3}}$

$ln x^{\sqrt[3]{4}}$

Solution

$\sqrt[3]{4} ln x$

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

${log}_{5} (4 x^{6} y^{4})$

${log}_{2} (3 x^{5} y^{3})$

Solution

${log}_{2} 3 + 5 {log}_{2} x + 3 {log}_{2} y$

${log}_{3} (\sqrt{2} x^{2})$

${log}_{5} (\sqrt[4]{21} y^{3})$

Solution

$\frac{1}{4} {log}_{5} 21 + 3 {log}_{5} y$

${log}_{3} \frac{x y^{2}}{z^{2}}$

${log}_{5} \frac{4 a b^{3} c^{4}}{d^{2}}$

Solution

${log}_{5} 4 + {log}_{5} a + 3 {log}_{5} b$
$+ 4 {log}_{5} c - 2 {log}_{5} d$

${log}_{4} \frac{\sqrt{x}}{16 y^{4}}$

${log}_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}$

Solution

$\frac{2}{3} {log}_{3} x - 3 - 4 {log}_{3} y$

${log}_{2} \frac{\sqrt{2 x + y^{2}}}{z^{2}}$

${log}_{3} \frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}}$

Solution

$\frac{1}{2} {log}_{3} (3 x + 2 y^{2}) - {log}_{3} 5 - 2 {log}_{3} z$

${log}_{2} \sqrt[4]{\frac{5 x^{3}}{2 y^{2} z^{4}}}$

${log}_{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}$

Solution

$\frac{1}{3} ({log}_{5} 3 + 2 {log}_{5} x - {log}_{5} 4$
$- 3 {log}_{5} y - {log}_{5} z)$

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

${log}_{6} 4 + {log}_{6} 9$

$log 4 + log 25$

Solution

${log}_{2} 80 - {log}_{2} 5$

${log}_{3} 36 - {log}_{3} 4$

Solution

${log}_{3} 4 + {log}_{3} (x + 1)$

${log}_{2} 5 - {log}_{2} (x - 1)$

Solution

${log}_{2} \frac{5}{x - 1}$

${log}_{7} 3 + {log}_{7} x - {log}_{7} y$

${log}_{5} 2 - {log}_{5} x - {log}_{5} y$

Solution

${log}_{5} \frac{2}{x y}$

$4 {log}_{2} x + 6 {log}_{2} y$

$6 {log}_{3} x + 9 {log}_{3} y$

Solution

${log}_{3} x^{6} y^{9}$

${log}_{3} (x^{2} - 1) - 2 {log}_{3} (x - 1)$

$log (x^{2} + 2 x + 1) - 2 log (x + 1)$

Solution

$4 log x - 2 log y - 3 log z$

$3 ln x + 4 ln y - 2 ln z$

Solution

$ln \frac{x^{3} y^{4}}{z^{2}}$

$\frac{1}{3} log x - 3 log (x + 1)$

$2 log (2 x + 3) + \frac{1}{2} log (x + 1)$

Solution

$log {(2 x + 3)}^{2} \cdot \sqrt{x + 1}$

Use the Change-of-Base Formula

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

${log}_{3} 42$

${log}_{5} 46$

Solution

$2.379$

${log}_{12} 87$

${log}_{15} 93$

Solution

$1.674$

${log}_{\sqrt{2}} 17$

${log}_{\sqrt{3}} 21$

Solution

$5.542$

Writing Exercises

Write the Product Property in your own words. Does it apply to each of the following? ${log}_{a} 5 x,$ ${log}_{a} (5 + x) .$ Why or why not?

Write the Power Property in your own words. Does it apply to each of the following? ${log}_{a} x^{p},$ ${({log}_{a} x)}^{r} .$ Why or why not?

Solution

Answers will vary.

Use an example to show that
$log (a + b) \neq log a + log b .$

Explain how to find the value of ${log}_{7} 15$ using your calculator.

Solution

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has three rows and four columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads use the properties of logarithms and use the change of base formula. The rest of the cells are blank.

ⓑ On a scale of $1 - 10,$ how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?