Precalculus by Richard Wright

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# 3-03 Properties of Logarithms

Summary: In this section, you will:

• Use properties of logarithms to expand logarithmic expressions.
• Use properties of logarithms to condense logarithmic expressions.
• Use the change-of-base formula to evaluate logarithms.
• Graph logarithmic functions.

SDA NAD Content Standards (2018): PC.5.3

In chemistry, pH is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. 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° 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 H+ is the concentration of hydrogen ion in the solution in mol/L.

pH = −log([H+])
$=log\left(\frac{1}{\left[{H}^{+}\right]}\right)$

The equivalence of −log([H+]) and $log\left(\frac{1}{\left[{H}^{+}\right]}\right)$ is one of the logarithm properties examined in this section.

## Properties of Logarithms

Remember that logarithms equal exponents. When two exponential expressions with the same base are multiplied, then the exponents are added. So when a logarithm is taken of two multiplied values, it can be rewritten as the sum of two separate logarithms. This is the product property.

logb (uv) = logb u + logb v

A similar property is the quotient property. When exponential expressions are divided, then the exponents are subtracted. So when a logarithm is taken of two divided values, it can be rewritten as the difference of two separate logarithms.

${log}_{b}\left(\frac{u}{v}\right)={log}_{b}u-{log}_{b}v$

When an exponential expression is given an additional exponent, the exponents are multiplied. The power property says that an exponent on a logarithm, it can be rewritten as multiplication.

logb un = n logb u

###### Properties of Logarithms
1. Product Property: logb (uv) = logb u + logb v
2. Quotient Property: ${log}_{b}\left(\frac{u}{v}\right)={log}_{b}u-{log}_{b}v$
3. Power Property: logb un = n logb u

#### Example 1: Use Properties of Exponents

Rewrite the expression in terms of log 2 and log 5.

1. log 10
2. log (5⁄2)
3. log 125
###### Solution
1. 10 = 2 ⋅ 5. Since this is multiplication, use the product property. log 10 = log 2 + log 5
2. This is division, so use the quotient property. log (5⁄2) = log 5 − log 2
3. 125 = 53, so use the power property. log 125 = log 53 = 3 log 5
##### Try It 1

Rewrite the expression in terms of ln 3 and ln 4.

1. $ln\left(\frac{3}{4}\right)$
2. ln 16

a. ln 3 − ln 4; b. 2 ln 4

### Expand and Condense Logarithmic Expressions

Separating expressions into several separate logarithmic expressions or combining logarithmic expressions together can be useful for solving logarithmic equations. Expanding expressions means to rewrite a single logarithmic expression into several expressions. Condensing means to do the opposite and rewrite several logarithmic expressions into a single logarithm.

#### Example 2: Expand a Logarithmic Expression

Expand log2 (3x5)

###### Solution

First use the product property since the exponent is only on the x and not on the 3.

log2 (3x5) = log2 3 + log2 x5

Now use the power property on the second term.

log2 3 + log2 x5 = log2 3 + 5 log2 x

##### Try It 2

Expand ${log}_{3}{\left(\frac{2}{x}\right)}^{4}$

4(log3 2 − log3 x)
Since the exponent is on both the 2 and x, the power property is first then the quotient property.

#### Example 3: Condense Logarithmic Expressions

Condense ln 2 + 4 ln y − ln x

###### Solution

Before the product or quotient properties can be used, the 4 needs to be moved from in front of its logarithm. Begin with the power property on the middle term.

ln 2 + 4 ln 3 − ln x = ln 2 + ln y4 − ln x

Now use the product and quotient properties. Any term that is subtracted will be in the denominator of the fraction.

ln 2 + ln y4 − ln x = $ln\left(\frac{2{y}^{4}}{x}\right)$

##### Try It 3

Condense log2 x − 3 log2 z

${log}_{2}\left(\frac{x}{{z}^{3}}\right)$

## Change-of-Base Formula

There is a reason why most calculators only have the common log and natural log buttons. The reason is that the base of the logarithm can be changed with a formula.

To derive the formula, call the logarithm we are trying to evaluate y.

logb x = y

Rewrite it as an exponential function.

x = by

Remember the Golden Rule of Algebra, what you do to one side of the equation, do to the other side. Take a logarithm of both sides. Any base is fine.

logc x = logc by

Use the power property on the right hand side, then solve for y.

logc x = y logc b
$\frac{{log}_{c}x}{{log}_{c}b}=y$

Replace y with the logarithm that it equals from the first step which results in the change-of-base formula.

${log}_{b}x=\frac{{log}_{c}x}{{log}_{c}b}$

###### Change-of-Base Formula

To evaluate any logarithm base b of x, use the following formula where c is any real number greater than zero. c is usually 10 or e.

${log}_{b}x=\frac{{log}_{c}x}{{log}_{c}b}$

#### Example 4: Evaluate a Logarithm using the Change-of-Base Formula

Evaluate log3 15.

###### Solution

b = 3 and x = 15. Plug those values into the formula letting c equal 10 or e so we can use a calculator. For now we will use 10.

${log}_{3}15=\frac{{log}_{10}15}{{log}_{10}3}=\frac{log15}{log3}\approx 2.465$

##### Try It 4

Evaluate log2 20.

4.322

### Graph Logarithmic Functions

Since logarithms are inverses of exponential functions, the graph of a logarithm is a reflection of an exponential function reflected over the line y = x.

The graph of a logarithmic function, g(x) = logb (xh) + k has several properties:

• Vertical asymptote at x = h
• x-intercept at (1, 0) if h = k = 0
• Domain: (h, ∞)
• Range: All Real Numbers
• Increasing from h to ∞

Logarithmic functions can be easily graphed by using the change-of-base formula to create a table of values. It is also important to graph the vertical asymptote.

###### Graph a Logarithmic Function

To graph a logarithmic function g(x) = logb (xh) + k

1. Identify h and graph the vertical asymptote x = h.
2. Rewrite the function in base 10 or e using the change-of-base formula.
3. Create a table of values.
4. Plot the points.
5. Draw a curve through the points and approaching the asymptote.

#### Example 5: Graph a Logarithmic Function

Graph f(x) = log2 (x + 1).

###### Solution

Compare f(x) with g(x) = logb (xh) + k. This gives h = −1, so the vertical asymptote is x = −1.

Use the change-of-base formula to rewrite the function with a base on the calculator.

$\begin{array}{l}f\left(x\right)={log}_{2}\left(x+1\right)\\ =\frac{{log}_{10}\left(x+1\right)}{{log}_{10}2}\\ =\frac{log\left(x+1\right)}{log2}\end{array}$

Create a table of values using x-values larger than h because that is the domain.

 x f(x) 0 1 2 3 4 5 6 0 1 1.585 2 2.322 2.585 2.807

When drawing the graph, graph the asymptote and make sure the curve approaches the asymptote.

##### Try It 5

Graph g(x) = log3 (x + 2) − 1.

##### Lesson Summary
###### Properties of Logarithms
1. Product Property: logb (uv) = logb u + logb v
2. Quotient Property: ${log}_{b}\left(\frac{u}{v}\right)={log}_{b}u-{log}_{b}v$
3. Power Property: logb un = n logb u

###### Change-of-Base Formula

To evaluate any logarithm base b of x, use the following formula where c is any real number greater than zero. c is usually 10 or e.

${log}_{b}x=\frac{{log}_{c}x}{{log}_{c}b}$

###### Graph a Logarithmic Function

To graph a logarithmic function g(x) = logb (xh) + k

1. Identify h and graph the vertical asymptote x = h.
2. Rewrite the function in base 10 or e using the change-of-base formula.
3. Create a table of values.
4. Plot the points.
5. Draw a curve through the points and approaching the asymptote.

## Practice Exercises

1. Which type of translation affect the domain of a logarithmic function?
2. Rewrite the expression in terms of log3 4 and log3 8.

3. log3 64
4. log3 2
5. Expand the following expressions.

6. log3 (3xy4)
7. $$\log \left(\frac{m}{3n^3}\right)$$
8. $$\ln \left(\frac{2p^3}{e^t ·q}\right)$$
9. Condense the following expressions.

10. ln x + 2 ln y
11. 2 log6 3 + log6 w − 4 log6 t
12. log2 7 − 2 log2 m − 4 log2 n
13. Use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to three decimal places.

14. log7 128
15. log½ 7.3
16. State the asymptote, domain, and range. Then sketch the graph of the indicated function.

17. f(x) = log2 (x − 3)
18. g(x) = log x + 1
19. h(x) = −2 ln (x − 3) + 1
20. Problem Solving

21. pH is a measure of active hydrogen ion in a solution. The formula pH = −log([H+]) is used to calculate pH where H+ is the hydrogen ion concentration in mol/L and pH is the pH level. What is the difference in pH level between battery acid with a hydrogen ion concentration of 1.58×10−1 mol/L and orange juice with a hydrogen ion concentration of 5.01×10−4 mol/L?
22. Mixed Review

23. (3-02) Rewrite as an exponential equation: ln x = 7.
24. (3-02) Evaluate without using a calculator: log3 (27) − 1.
25. (3-01) For the following exercises, graph the function. Then state the domain, range, asymptote, and whether it is exponential growth, decay, or neither: f(x) = ½ex + 1
26. (2-09) Solve by graphing: 2x2 − 5x − 3 ≤ 0.
27. (2-06) Solve x3 − 7x2 + 17x − 15 = 0.

1. Shifting the function right or left will affect its domain.
2. 3 log3 4
3. log3 8 − log3 4
4. 1 + log3 (x) + 4 log3 (y)
5. log m − log 3 − 3 log n
6. ln 2 + 3 ln pt − ln q
7. ln xy2
8. $$\log_{6} \left(\frac{9w}{t^4}\right)$$
9. $$\log_{2} \left(\frac{7}{m^2 n^4}\right)$$
10. 2.493
11. −2.868
12. VA: x = 3; Domain: (3, ∞); Range: (−∞, ∞);
13. VA: x = 0; Domain: (0, ∞); Range: (−∞, ∞);
14. VA: x = 3; Domain: (3, ∞); Range: (−∞, ∞);
15. 2.499
16. e7 = x
17. 2
18. ; Domain: (−∞, ∞); Range: (1, ∞); HA: y = 1; Exponential growth
19. [−½, 3]
20. 3, 2 + i, 2 − i