10.4 Use the Properties of Logarithms

Learning objectives

Learning objectives

• Use the properties of logarithms

• Use the Change of Base Formula

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 ${\mathrm{log}}_{a}1=0.$ Also, since ${a}^{1}=a,$ we get ${\mathrm{log}}_{a}a=1.$

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.

Example 10.28

Solution

ⓐ

$\,{\mathrm{log}}_{8}1$

Use the property, ${\mathrm{log}}_{a}1=0$. · $\,0\,{\mathrm{log}}_{8}1=0$

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

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

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

The logarithmic equation ${\mathrm{log}}_{a}{a}^{x}=x$ 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.

Example 10.29

Solution

ⓐ

$\,{4}^{{\mathrm{log}}_{4}9}$

Use the property, ${a}^{{\mathrm{log}}_{a}x}=x$. · $\,9\,{4}^{{\mathrm{log}}_{4}9}=9$

ⓑ

$\,{\mathrm{log}}_{3}{3}^{5}$

Use the property, ${a}^{{\mathrm{log}}_{a}x}=x$. · $\,5\,{\mathrm{log}}_{3}{3}^{5}=5$

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, ${\mathrm{log}}_{a}M\cdot{}N={\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N$ 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.

Example 10.30

Solution

ⓐ

$\,{\mathrm{log}}_{3}7x$

Use the Product Property, ${\mathrm{log}}_{a}(M\cdot{}N)={\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N$. · $\,{\mathrm{log}}_{3}7+{\mathrm{log}}_{3}x$

$\,{\mathrm{log}}_{3}7x={\mathrm{log}}_{3}7+{\mathrm{log}}_{3}x$

ⓑ

$\,{\mathrm{log}}_{4}64xy$

Use the Product Property, ${\mathrm{log}}_{a}(M\cdot{}N)={\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N$. · $\,{\mathrm{log}}_{4}64+{\mathrm{log}}_{4}x+{\mathrm{log}}_{4}y$ Simplify by evaluating ${\mathrm{log}}_{4}64$. · $\,3+{\mathrm{log}}_{4}x+{\mathrm{log}}_{4}y$

$\,{\mathrm{log}}_{4}64xy=3+{\mathrm{log}}_{4}x+{\mathrm{log}}_{4}y$

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, ${\mathrm{log}}_{a}\frac{M}{N}={\mathrm{log}}_{a}M-{\mathrm{log}}_{a}N$ tells us to take the log of a quotient, we subtract the log of the numerator and denominator.

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

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

Example 10.31

Solution

ⓐ

$\,{\mathrm{log}}_{5}\frac{5}{7}$

Use the Quotient Property, ${\mathrm{log}}_{a}\frac{M}{N}={\mathrm{log}}_{a}M-{\mathrm{log}}_{a}N$. · $\,{\mathrm{log}}_{5}5-{\mathrm{log}}_{5}7$ Simplify. · $\,1-{\mathrm{log}}_{5}7$

$\,{\mathrm{log}}_{5}\frac{5}{7}=1-{\mathrm{log}}_{5}7$

ⓑ

$\,\mathrm{log}\frac{x}{100}$

Use the Quotient Property, ${\mathrm{log}}_{a}\frac{M}{N}={\mathrm{log}}_{a}M-{\mathrm{log}}_{a}N$. · $\,\mathrm{log}\,x-\mathrm{log}100$ Simplify. · $\,\mathrm{log}\,x-2$

$\,\mathrm{log}\frac{x}{100}=\mathrm{log}\,x-2$

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, ${\mathrm{log}}_{a}{M}^{p}=p\,{\mathrm{log}}_{a}M$ 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.

Example 10.32

Solution

ⓐ

$\,{\mathrm{log}}_{5}{4}^{3}$

Use the Power Property, ${\mathrm{log}}_{a}{M}^{p}=p\,{\mathrm{log}}_{a}M$. · $\,3{\mathrm{log}}_{5}4$

$\,{\mathrm{log}}_{5}{4}^{3}=3{\mathrm{log}}_{5}4$

ⓑ

$\,\mathrm{log}{x}^{10}$

Use the Power Property, ${\mathrm{log}}_{a}{M}^{p}=p\,{\mathrm{log}}_{a}M$. · $\,10\,\mathrm{log}\,x$

$\,\mathrm{log}{x}^{10}=10\mathrm{log}\,x$

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.

Example 10.33

Solution

$\,{\mathrm{log}}_{4}(2{x}^{3}{y}^{2})$

Use the Product Property, ${\mathrm{log}}_{a}M\cdot{}N={\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N$. · $\,{\mathrm{log}}_{4}2+{\mathrm{log}}_{4}{x}^{3}+{\mathrm{log}}_{4}{y}^{2}$ Use the Power Property, ${\mathrm{log}}_{a}{M}^{p}=p\,{\mathrm{log}}_{a}M$, on the last two terms. · $\,{\mathrm{log}}_{4}2+3{\mathrm{log}}_{4}x+2{\mathrm{log}}_{4}y$ Simplify. · $\,\frac{1}{2}+3{\mathrm{log}}_{4}x+2{\mathrm{log}}_{4}y$

$\,{\mathrm{log}}_{4}(2{x}^{3}{y}^{2})=\frac{1}{2}+3{\mathrm{log}}_{4}x+2{\mathrm{log}}_{4}y$

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

Example 10.34

Solution

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

Rewrite the radical with a rational exponent. · $\,{\mathrm{log}}_{2}{(\frac{{x}^{3}}{3{y}^{2}z})}^{\frac{1}{4}}$ Use the Power Property, ${\mathrm{log}}_{a}{M}^{p}=p\,{\mathrm{log}}_{a}M$. · $\,\frac{1}{4}{\mathrm{log}}_{2}(\frac{{x}^{3}}{3{y}^{2}z})$ Use the Quotient Property, ${\mathrm{log}}_{a}M\cdot{}N={\mathrm{log}}_{a}M-{\mathrm{log}}_{a}N$. · $\,\frac{1}{4}({\mathrm{log}}_{2}({x}^{3})-{\mathrm{log}}_{2}(3{y}^{2}z))$ Use the Product Property, ${\mathrm{log}}_{a}M\cdot{}N={\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N$, in the second term. · $\,\frac{1}{4}({\mathrm{log}}_{2}({x}^{3})-({\mathrm{log}}_{2}3+{\mathrm{log}}_{2}{y}^{2}+{\mathrm{log}}_{2}z))$ Use the Power Property, ${\mathrm{log}}_{a}{M}^{p}=p\,{\mathrm{log}}_{a}M$, inside the parentheses. · $\,\frac{1}{4}(3{\mathrm{log}}_{2}x-({\mathrm{log}}_{2}3+2{\mathrm{log}}_{2}y+{\mathrm{log}}_{2}z))$ Simplify by distributing. · $\,\frac{1}{4}(3{\mathrm{log}}_{2}x-{\mathrm{log}}_{2}3-2{\mathrm{log}}_{2}y-{\mathrm{log}}_{2}z)$

${\mathrm{log}}_{2}\sqrt[4]{\frac{{x}^{3}}{3{y}^{2}z}}=\frac{1}{4}(3{\mathrm{log}}_{2}x-{\mathrm{log}}_{2}3-2{\mathrm{log}}_{2}y-{\mathrm{log}}_{2}z)$

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.

Example 10.35

Solution

The log expressions all have the same base, 4. · $\,{\mathrm{log}}_{4}3+{\mathrm{log}}_{4}x-{\mathrm{log}}_{4}y$ The first two terms are added, so we use the Product Property, ${\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N={\mathrm{log}}_{a}M\cdot{}N$. · $\,{\mathrm{log}}_{4}3x-{\mathrm{log}}_{4}y$ Since the logs are subtracted, we use the Quotient Property, ${\mathrm{log}}_{a}M-{\mathrm{log}}_{a}N={\mathrm{log}}_{a}\frac{M}{N}$. · $\,{\mathrm{log}}_{4}\frac{3x}{y}$

$\,{\mathrm{log}}_{4}3+{\mathrm{log}}_{4}x-{\mathrm{log}}_{4}y={\mathrm{log}}_{4}\frac{3x}{y}$

Example 10.36

Solution

The log expressions have the same base, 3. · $\,2{\mathrm{log}}_{3}x+4{\mathrm{log}}_{3}(x+1)$ Use the Power Property, ${\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N={\mathrm{log}}_{a}M\cdot{}N$. · $\,{\mathrm{log}}_{3}{x}^{2}+{\mathrm{log}}_{3}{(x+1)}^{4}$ The terms are added, so we use the Product Property, ${\mathrm{log}}_{a}M+{\mathrm{log}}_{a}N={\mathrm{log}}_{a}M\cdot{}N$. · $\,{\mathrm{log}}_{3}{x}^{2}{(x+1)}^{4}$

$\,2{\mathrm{log}}_{3}x+4{\mathrm{log}}_{3}(x+1)={\mathrm{log}}_{3}{x}^{2}{(x+1)}^{4}$

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.

Suppose we want to evaluate ${\mathrm{log}}_{a}M$. · $\,{\mathrm{log}}_{a}M\,$ Let $y={\mathrm{log}}_{a}M$. · $\,y\,=\,{\mathrm{log}}_{a}M$ Rewrite the expression in exponential form. · $\,{a}^{y}\,=\,M\,$ Take the ${\mathrm{log}}_{b}$ of each side. · $\,{\mathrm{log}}_{b}{a}^{y}\,=\,{\mathrm{log}}_{b}M$ Use the Power Property. · $\,y{\mathrm{log}}_{b}a\,=\,{\mathrm{log}}_{b}M$ Solve for $y$. · $\,y\,=\,\frac{{\mathrm{log}}_{b}M}{{\mathrm{log}}_{b}a}$ Substitute $y={\mathrm{log}}_{a}M$. · $\,{\mathrm{log}}_{a}M\,=\,\frac{{\mathrm{log}}_{b}M}{{\mathrm{log}}_{b}a}$

The Change-of-Base Formula introduces a new base $b.$ This can be any base b we want where $b>0,b\ne{}1.$ 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 $\mathrm{(≈)}$.

Example 10.37

Solution

Section 10.4 Exercises

Section 10.4 Exercises

Use the Properties of Logarithms

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

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

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

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

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

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

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.

Writing Exercises

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

ⓑ 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?

1. Evaluate using the properties of logarithms: ⓐ ${\mathrm{log}}_{13}1$ ⓑ ${\mathrm{log}}_{9}9.$

2. Evaluate using the properties of logarithms: ⓐ ${\mathrm{log}}_{5}1$ ⓑ ${\mathrm{log}}_{7}7.$

3. Evaluate using the properties of logarithms: ⓐ ${5}^{{\mathrm{log}}_{5}15}$ ⓑ ${\mathrm{log}}_{7}{7}^{4}.$

4. Evaluate using the properties of logarithms: ⓐ ${2}^{{\mathrm{log}}_{2}8}$ ⓑ ${\mathrm{log}}_{2}{2}^{15}.$

1. Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

ⓐ ${\mathrm{log}}_{3}3x$ ⓑ ${\mathrm{log}}_{2}8xy$

2. Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

ⓐ ${\mathrm{log}}_{9}9x$ ⓑ ${\mathrm{log}}_{3}27xy$

3. Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

ⓐ ${\mathrm{log}}_{4}\frac{3}{4}$ ⓑ $\mathrm{log}\frac{x}{1000}$

4. Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.

ⓐ ${\mathrm{log}}_{2}\frac{5}{4}$ ⓑ $\mathrm{log}\frac{10}{y}$

1. Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

ⓐ ${\mathrm{log}}_{7}{5}^{4}$ ⓑ $\mathrm{log}{x}^{100}$

2. Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.

ⓐ ${\mathrm{log}}_{2}{3}^{7}$ ⓑ $\mathrm{log}{x}^{20}$

3. Use the Properties of Logarithms to expand the logarithm ${\mathrm{log}}_{2}(5{x}^{4}{y}^{2})$. Simplify, if possible.

4. Use the Properties of Logarithms to expand the logarithm ${\mathrm{log}}_{3}(7{x}^{5}{y}^{3})$. Simplify, if possible.

1. Use the Properties of Logarithms to expand the logarithm ${\mathrm{log}}_{4}\sqrt[5]{\frac{{x}^{4}}{2{y}^{3}{z}^{2}}}$. Simplify, if possible.

2. Use the Properties of Logarithms to expand the logarithm ${\mathrm{log}}_{3}\sqrt[3]{\frac{{x}^{2}}{5{y}^{}z}}$. Simplify, if possible.

3. Use the Properties of Logarithms to condense the logarithm ${\mathrm{log}}_{2}5+{\mathrm{log}}_{2}x-{\mathrm{log}}_{2}y$. Simplify, if possible.

4. Use the Properties of Logarithms to condense the logarithm ${\mathrm{log}}_{3}6-{\mathrm{log}}_{3}x-{\mathrm{log}}_{3}y$. Simplify, if possible.

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

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

3. Rounding to three decimal places, approximate ${\mathrm{log}}_{3}42.$

4. Rounding to three decimal places, approximate ${\mathrm{log}}_{5}46.$

1. ⓐ ${\mathrm{log}}_{12}1$ ⓑ $\mathrm{ln}\,e$

2. ⓐ ${5}^{{\mathrm{log}}_{5}10}$ ⓑ ${\mathrm{log}}_{4}{4}^{10}$

3. ⓐ ${6}^{{\mathrm{log}}_{6}15}$ ⓑ ${\mathrm{log}}_{8}{8}^{-4}$

4. ⓐ ${10}^{\mathrm{log}\sqrt{3}}$ ⓑ $\mathrm{log}{10}^{-1}$

1. ⓐ ${e}^{\mathrm{ln}3}$ ⓑ $\mathrm{ln}\,{e}^{7}$

2. ${\mathrm{log}}_{5}8y$

3. ${\mathrm{log}}_{3}81xy$

4. $\mathrm{log}1000y$

1. ${\mathrm{log}}_{6}\frac{5}{6}$

2. ${\mathrm{log}}_{5}\frac{125}{x}$

3. $\mathrm{log}\frac{10,000}{y}$

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

1. ${\mathrm{log}}_{2}{x}^{5}$

2. $\mathrm{log}{x}^{-3}$

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

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

1. ${\mathrm{log}}_{2}(3{x}^{5}{y}^{3})$

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

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

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

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

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

3. ${\mathrm{log}}_{2}5-{\mathrm{log}}_{2}(x-1)$

4. ${\mathrm{log}}_{5}2-{\mathrm{log}}_{5}x-{\mathrm{log}}_{5}y$

1. $6{\mathrm{log}}_{3}x+9{\mathrm{log}}_{3}y$

2. $3\mathrm{ln}\,x+4\mathrm{ln}\,y-2\mathrm{ln}\,z$

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

4. ${\mathrm{log}}_{5}46$

1. ${\mathrm{log}}_{15}93$

2. ${\mathrm{log}}_{\sqrt{3}}21$

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

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