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## 7.4 Properties of the Logarithm

### Learning Objectives

1. Apply the inverse properties of the logarithm.
2. Expand logarithms using the product, quotient, and power rule for logarithms.
3. Combine logarithms into a single logarithm with coefficient 1.

## Logarithms and Their Inverse Properties

Recall the definition of the base-b logarithm: given $b>0$ where $b≠1$,

$y=logb x if and only if x=by$

Use this definition to convert logarithms to exponential form. Doing this, we can derive a few properties:

$logb 1=0 because b0=1logb b=1 because b1=blogb (1b)=−1 because b−1=1b$

### Example 1

Evaluate:

1. $log 1$
2. $ln e$
3. $log5 (15)$

Solution:

1. When the base is not written, it is assumed to be 10. This is the common logarithm,

$log 1=log10 1=0$

2. The natural logarithm, by definition, has base e,

$ln e=loge e=1$

3. Because $5−1=15$ we have,

$log5 (15)=−1$

Furthermore, consider fractional bases of the form $1/b$ where $b>1.$

$log1/b b=−1 because (1b)−1=1−1b−1=b1=b$

### Example 2

Evaluate:

1. $log1/4 4$
2. $log2/3 (32)$

Solution:

1. $log1/4 4=−1$ because $(14)−1=4$
2. $log2/3 (32)=−1$ because $(23)−1=32$

Given an exponential function defined by $f(x)=bx$, where $b>0$ and $b≠1$, its inverse is the base-b logarithm, $f−1(x)=logb x.$ And because $f(f−1(x))=x$ and $f−1(f(x))=x$, we have the following inverse properties of the logarithmGiven $b>0$ we have $logb bx=x$ and $blogb x=x$ when $x>0.$:

$f−1(f(x))=logb bx=xandf(f−1(x))=blogb x=x ,x>0$

Since $f−1(x)=logb x$ has a domain consisting of positive values $(0,∞)$, the property $blogb x=x$ is restricted to values where $x>0.$

### Example 3

Evaluate:

1. $log5 625$
2. $5log5 3$
3. $eln 5$

Solution:

Apply the inverse properties of the logarithm.

1. $log5 625=log5 54=4$
2. $5log5 3=3$
3. $eln 5=5$

In summary, when $b>0$ and $b≠1$, we have the following properties:

 $logb 1=0$ $logb b=1$ $log1/b b=−1$ $logb (1b)=−1$ $logb bx=x$ $blogb x=x$, $x>0$

Try this! Evaluate: $log 0.00001$

## Product, Quotient, and Power Properties of Logarithms

In this section, three very important properties of the logarithm are developed. These properties will allow us to expand our ability to solve many more equations. We begin by assigning u and v to the following logarithms and then write them in exponential form:

$logb x=u ⇒ bu=xlogb y=v ⇒ bv=y$

Substitute $x=bu$ and $y=bv$ into the logarithm of a product $logb (xy)$ and the logarithm of a quotient $logb (xy).$ Then simplify using the rules of exponents and the inverse properties of the logarithm.

Logarithm of a Product

Logarithm of a Quotient

$logb (xy)=logb (bubv)=logb bu+v=u+v=logb x+logb y$

$logb (xy)=logb (bubv)=logb bu−v=u−v=logb x−logb y$

This gives us two essential properties: the product property of logarithms$logb (xy)=logb x+logb y;$ the logarithm of a product is equal to the sum of the logarithm of the factors.,

$logb (xy)=logb x+logb y$

and the quotient property of logarithms$logb (xy)=logb x−logb y;$ the logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator.,

$logb (xy)=logb x−logb y$

In words, the logarithm of a product is equal to the sum of the logarithm of the factors. Similarly, the logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator.

### Example 4

Write as a sum: $log2 (8x).$

Solution:

Apply the product property of logarithms and then simplify.

$log2 (8x)=log2 8+log2 x=log2 23+log2 x=3+log2 x$

Answer: $3+log2 x$

### Example 5

Write as a difference: $log (x10)$.

Solution:

Apply the quotient property of logarithms and then simplify.

$log (x10)=log x−log 10=log x−1$

Answer: $log x−1$

Next we begin with $logb x=u$ and rewrite it in exponential form. After raising both sides to the nth power, convert back to logarithmic form, and then back substitute.

$logb x=u ⇒ bu=x(bu)n=(x )nlogb xn=nu ⇐ bnu=xnlogb xn=nlogb x$

This leads us to the power property of logarithms$logb xn=nlogb x$; the logarithm of a quantity raised to a power is equal to that power times the logarithm of the quantity.,

$logb xn=nlogb x$

In words, the logarithm of a quantity raised to a power is equal to that power times the logarithm of the quantity.

### Example 6

Write as a product:

1. $log2 x4$
2. $log5 (x).$

Solution:

1. Apply the power property of logarithms.

$log2 x4=4log2 x$

2. Recall that a square root can be expressed using rational exponents, $x=x1/2.$ Make this replacement and then apply the power property of logarithms.

$log5 (x)=log5 x1/2=12log5 x$

In summary,

 Product property of logarithms $logb (xy)=logb x+logb y$ Quotient property of logarithms $logb (xy)=logb x−logb y$ Power property of logarithms $logb xn=nlogb x$

We can use these properties to expand logarithms involving products, quotients, and powers using sums, differences and coefficients. A logarithmic expression is completely expanded when the properties of the logarithm can no further be applied.

$Caution:$ It is important to point out the following:

$log (xy)≠log x⋅log y and log (xy)≠log xlog y$

### Example 7

Expand completely: $ln (2x3).$

Solution:

Recall that the natural logarithm is a logarithm base e, $ln x=loge x.$ Therefore, all of the properties of the logarithm apply.

$ln (2x3)=ln 2+ln x3 Product rule for logarithms=ln 2+3ln x Power rule for logarithms$

Answer: $ln 2+3ln x$

### Example 8

Expand completely: $log 10xy23.$

Solution:

Begin by rewriting the cube root using the rational exponent $13$ and then apply the properties of the logarithm.

$log 10xy23=log (10xy2)1/3=13log (10xy2)=13(log 10+log x+log y2)=13(1+log x+2log y)=13+13log x+23log y$

Answer: $13+13log x+23log y$

### Example 9

Expand completely: $log2 ( (x+1)25y)$.

Solution:

When applying the product property to the denominator, take care to distribute the negative obtained from applying the quotient property.

$log2 ( (x+1)25y)=log2 (x+1)2−log2 (5y)=log2 (x+1)2−(log2 5+log2 y) Distribute.=log2 (x+1)2−log2 5−log2 y=2log2 (x+1)−log2 5−log2 y$

Answer: $2log2 (x+1)−log2 5−log2 y$

Caution: There is no rule that allows us to expand the logarithm of a sum or difference. In other words,

$log (x±y)≠log x±log y$

Try this! Expand completely: $ln (5y4x)$.

Answer: $ln 5+4ln y−12ln x$

### Example 10

Given that $log2 x=a$, $log2 y=b$, and that $log2 z=c$, write the following in terms of a, b and c:

a. $log2 (8x2y)$

b. $log2 (2x4z)$

Solution:

1. Begin by expanding using sums and coefficients and then replace a and b with the appropriate logarithm.

$log2 (8x2y)=log2 8+log2 x2+log2 y=log2 8+2log2 x+log2 y=3+2a+b$

2. Expand and then replace a, b, and c where appropriate.

$log2 (2x4z)=log2 (2x4)−log2 z1/2=log2 2+log2 x4−log2 z1/2=log2 2+4log2 x−12log2 z=1+4a−12b$

Next we will condense logarithmic expressions. As we will see, it is important to be able to combine an expression involving logarithms into a single logarithm with coefficient 1. This will be one of the first steps when solving logarithmic equations.

### Example 11

Write as a single logarithm with coefficient $1: 3log3 x−log3 y+2log3 5.$

Solution:

Begin by rewriting all of the logarithmic terms with coefficient 1. Use the power rule to do this. Then use the product and quotient rules to simplify further.

$3log3x−log3y+2log35={log3x3−log3y}+log352 quotient property={log3(x3y)+log325} product property=log3(x3y⋅25)=log3(25x3y)$

Answer: $log3 (25x3y)$

### Example 12

Write as a single logarithm with coefficient $1: 12ln x−3ln y−ln z.$

Solution:

Begin by writing the coefficients of the logarithms as powers of their argument, after which we will apply the quotient rule twice working from left to right.

$12ln x−3ln y−ln z=ln x1/2−ln y3−ln z=ln (x1/2y3)−ln z=ln (x1/2y3÷z)=ln (x1/2y3⋅1z)=ln (x1/2y3z) or =ln (xy3z)$

Answer: $ln (xy3z)$

Try this! Write as a single logarithm with coefficient $1: 3log (x+y)−6log z+2log 5.$

Answer: $log (25(x+y)3z6)$

### Key Takeaways

• Given any base $b>0$ and $b≠1$, we can say that $logb 1=0$, $logb b=1$, $log1/b b=−1$ and that $logb (1b)=−1.$
• The inverse properties of the logarithm are $logb bx=x$ and $blogb x=x$ where $x>0.$
• The product property of the logarithm allows us to write a product as a sum: $logb (xy)=logb x+logb y.$
• The quotient property of the logarithm allows us to write a quotient as a difference: $logb (xy)=logb x−logb y.$
• The power property of the logarithm allows us to write exponents as coefficients: $logb xn=nlogb x.$
• Since the natural logarithm is a base-e logarithm, $ln x=loge x$, all of the properties of the logarithm apply to it.
• We can use the properties of the logarithm to expand logarithmic expressions using sums, differences, and coefficients. A logarithmic expression is completely expanded when the properties of the logarithm can no further be applied.
• We can use the properties of the logarithm to combine expressions involving logarithms into a single logarithm with coefficient 1. This is an essential skill to be learned in this chapter.

### Part A: Logarithms and Their Inverse Properties

Evaluate:

1. $log7 1$

2. $log1/2 2$

3. $log 1014$

4. $log 10−23$

5. $log3 310$

6. $log6 6$

7. $ln e7$

8. $ln (1e)$
9. $log1/2 (12)$
10. $log1/5 5$

11. $log3/4 (43)$

12. $log2/3 1$

13. $2log2 100$

14. $3log3 1$

15. $10log 18$

16. $eln 23$

17. $eln x2$

18. $eln ex$

Find a:

1. $ln a=1$

2. $log a=−1$

3. $log9 a=−1$

4. $log12 a=1$

5. $log2 a=5$

6. $log a=13$

7. $2a=7$

8. $ea=23$

9. $loga 45=5$

10. $loga 10=1$

### Part B: Product, Quotient, and Power Properties of Logarithms

Expand completely.

1. $log4 (xy)$

2. $log (6x)$

3. $log3 (9x2)$

4. $log2 (32x7)$

5. $ln (3y2)$

6. $log (100x2)$

7. $log2 (xy2)$
8. $log5 (25x)$
9. $log (10x2y3)$

10. $log2 (2x4y5)$

11. $log3 (x3yz2)$
12. $log (xy3z2)$
13. $log5 (1x2yz)$
14. $log4 (116x2z3)$
15. $log6 [36(x+y)4]$

16. $ln [e4(x−y)3]$

17. $log7 (2xy)$

18. $ln (2xy)$

19. $log3 (x2y3z)$
20. $log (2(x+y)3z2)$
21. $log (100x3(y+10)3)$
22. $log7 (x(y+z)35)$
23. $log5 (x3yz23)$
24. $log (x2y3z25)$

Given $log3 x=a$, $log3 y=b$, and $log3 z=c$, write the following logarithms in terms of a, b, and c.

1. $log3 (27x2y3z)$
2. $log3 (xy3z)$
3. $log3 (9x2yz3)$
4. $log3 (x3yz2)$

Given $logb 2=0.43$, $logb 3=0.68$, and $logb 7=1.21$, calculate the following. (Hint: Expand using sums, differences, and quotients of the factors 2, 3, and 7.)

1. $logb 42$

2. $logb (36)$

3. $logb (289)$

4. $logb 21$

Expand using the properties of the logarithm and then approximate using a calculator to the nearest tenth.

1. $log (3.10×1025)$

2. $log (1.40×10−33)$

3. $ln (6.2e−15)$

4. $ln (1.4e22)$

Write as a single logarithm with coefficient 1.

1. $log x+log y$

2. $log3 x−log3 y$

3. $log2 5+2log2 x+log2 y$

4. $log3 4+3log3 x+12log3 y$

5. $3log2 x−2log2 y+12log2 z$

6. $4log x−log y−log 2$

7. $log 5+3log (x+y)$

8. $4log5 (x+5)+log5 y$

9. $ln x−6ln y+ln z$

10. $log3 x−2log3 y+5log3 z$

11. $7log x−log y−2log z$

12. $2ln x−3ln y−ln z$

13. $23log3 x−12(log3 y+log3 z)$

14. $15(log7 x+2log7 y)−2log7 (z+1)$

15. $1+log2 x−12log2 y$

16. $2−3log3 x+13log3 y$

17. $13log2 x+23log2 y$

18. $−2log5 x+35log5 y$

19. $−ln 2+2ln (x+y)−ln z$

20. $−3ln (x−y)−ln z+ln 5$

21. $13(ln x+2ln y)−(3ln 2+ln z)$

22. $4log 2+23log x−4log (y+z)$

23. $log2 3−2log2 x+12log2 y−4log2 z$

24. $2log5 4−log5 x−3log5 y+23log5 z$

Express as a single logarithm and simplify.

1. $log (x+1)+log (x−1)$

2. $log2 (x+2)+log2 (x+1)$

3. $ln (x2+2x+1)−ln (x+1)$

4. $ln (x2−9)−ln (x+3)$

5. $log5 (x3−8)−log5 (x−2)$

6. $log3 (x3+1)−log3 (x+1)$

7. $log x+log (x+5)−log (x2−25)$

8. $log (2x+1)+log (x−3)−log (2x2−5x−3)$

1. 0

2. 14

3. 10

4. 7

5. 1

6. −1

7. 100

8. 18

9. $x2$

10. $e$

11. $19$

12. $25=32$

13. $log2 7$

14. 4

1. $log4 x+log4 y$

2. $2+2log3 x$

3. $ln 3+2ln y$

4. $log2 x−2log2 y$

5. $1+2log x+3log y$

6. $3log3 x−log3 y−2log3 z$

7. $−2log5 x−log5 y−log5 z$

8. $2+4log6 (x+y)$

9. $log7 2+12log7 x+12log7 y$

10. $2log3 x+13log3 y−log3 z$

11. $2+3log x−3log (y+10)$

12. $3log5 x−13log5 y−23log5 z$

13. $3+2a+3b+c$

14. $2+2a+b−3c$

15. 2.32

16. 0.71

17. $log (3.1)+25≈25.5$

18. $ln (6.2)−15≈−13.2$

19. $log (xy)$

20. $log2 (5x2y)$

21. $log2 (x3zy2)$
22. $log [5(x+y)3]$

23. $ln (xzy6)$
24. $log (x7yz2)$
25. $log3 (x23yz)$
26. $log2 (2xy)$
27. $log2 (xy23)$
28. $ln ((x+y)22z)$
29. $ln (xy238z)$
30. $log2 (3yx2z4)$
31. $log (x2−1)$

32. $ln (x+1)$

33. $log5 (x2+2x+4)$

34. $log (xx−5)$