Free Printable Logarithms

1.

Which property of logarithms is demonstrated below:

\log_920\ =\ \frac{\log20}{\log9}

a)

Product property

b)

Quotient property

c)

Power property

d)

Change of Base Property

2.

Rewrite log_b(xⁿ)

a)

nlog_bx

b)

(log_bx)ⁿ

c)

xⁿlog_bx

d)

log_b(xⁿ)

3.

Use the properties of logarithms to retwrite

a)

A

b)

B

c)

C

d)

D

4.

Rewrite as a single logarithm:

\log3\ +\ \log7

a)

log 10

b)

log 21

c)

log 3/7

d)

log 3/log 7

5.

Rewrite as a single logarithm:

\log_260\ -\ \log_210

a)

$\log_26$

b)

$\log_250$

c)

$\frac{\log_260}{\log_210}$

d)

$\log_270$

6.

Use the properties of logarithms to rewrite as the sum of two logarithms:

\log55

a)

log 40 + log 15

b)

log 50 + log 5

c)

log 11 + log 5

7.

Which of the logarithms below is equivalent to the following:
$2\log12$

a)

log 10

b)

log 6

c)

log 24

d)

log 144

8.

Use the change of base property to rewrite as a single logarithm:

$\frac{\log15}{\log3}$

a)

$\log\ \frac{1}{5}$

b)

$\log5$

c)

$\log_{15}3$

d)

$\log_315$

9.

True or False:

log 12 - log 4 = log 8

a)

True

b)

False

10.

$-2\log3=\log\frac{1}{9}$

True or False:

a)

True

b)

False

11.

Expand $\log_6\left(\frac{5x^3}{y}\right)$

a)

log₆5x³-log₆y

b)

log₆5+log₆x³-log₆y

c)

log₆5+3log₆x-log₆y

12.

Expand: $\log_6\left(\frac{5}{4y}\right)$

a)

$\log_65\ +\ \log_64\ +\ \log_6y$

b)

$\log_65\ -\ \log_64\ +\ \log_6y$

c)

$\log_65\ -\ \log_64\ -\ \log_6y$

d)

$\log_65\ -\ \log_64y$

13.

Condense: $\log\left(5x+2\right)\ -\ \log3\ -\ \log x$

a)

$\log\left(\frac{3x}{5x\ +\ 2}\right)$

b)

$\log\left(\frac{5x\ +\ 2}{3x}\right)$

c)

$\log\left(3x\left(5x+2\right)\right)$

d)

$\log\left(\frac{5x\ +\ 2}{3\ -\ x}\right)$

14.

Condense: $\frac{1}{2}\log_3x\ +\ 5\log_3y$

a)

$\log\left(\sqrt{x}y^5\right)$

b)

$\log_3\left(\sqrt{x}y^5\right)$

c)

$\log_3\left(5\sqrt{x}y\right)$

d)

$\log_3\left(xy\right)^{\frac{5}{2}}$

15.

Condense: $3\ln x\ +\ 8\ln y$

a)

$\ln\left(\frac{x^3}{y^8}\right)$

b)

$\ln\left(x^3y^8\right)$

c)

$24\ln\left(xy\right)$

d)

$11\ln\left(xy\right)$

16.

Evaluate using properties of logarithms (no calculator): $3\ln e\ \ +\ 2\ln1$

a)

3

b)

4

c)

5

d)

e

17.

Evaluate using properties of logarithms (no calculator): $\log_5100\ -\ 2\log_52$

a)

50

b)

25

c)

20

d)

2

18.

Log with a base "e" (log_e) is the same thing as...

a)

"e"

b)

Natural Logarithm (LN)

c)

Common Logarithm (Log)

d)

Natural Log, base "e"
LN_e

19.

Expand: $\log_9\sqrt{23}$

a)

$\frac{1}{2}\log_923$

b)

$\log_9\left(\frac{1}{2}\right)-\log_923$

c)

$\log_9\sqrt{2}+\log_9\sqrt{3}$

d)

$\log_9\ \left(\frac{1}{2}\right)+\log_923$

20.

Expand: $\log_74\sqrt{d}$

a)

$\log_7\ 4\ -\ \frac{1}{2}\log_7\ d$

b)

$4\log_7d\ +\log_7\ \frac{1}{2}$

c)

$\log_7\ 4+\frac{1}{2}\log_7\ d$

d)

$\frac{1}{2}\log4d$

21.

Approximate its value to the nearest ten-thousandths using change of base formula. log₃7

a)

2.7712

b)

1.7712

c)

2

d)

1.7718

22.

Approximate the value to the nearest ten-thousandths using the change of base formula. log₂35

a)

1.5592

b)

5.9187

c)

5.1293

d)

1.8990

23.

Solve the equation. 3^x = 243

a)

x = 3

b)

x = 1

c)

x = 10

d)

x = 5

24.

7^y = 15

a)

2.8970

b)

1.3917

c)

1.8956

d)

5.2342

25.

5^3b = 106

a)

2.9605

b)

1.9659

c)

0.9659

d)

2.9638

26.

6^x+2 = 18

a)

0.3869

b)

-0.3869

c)

1.3869

d)

-1.7594

27.

7^3x-1 = 21

a)

x = 1.5645

b)

x = 1.8549

c)

x = 0.8549

d)

x = .7741

28.

5^{w + 3}= 17

a)

1.2379

b)

-1.2396

c)

2.9845

d)

-2.2396

29.

2.4^{x + 4} = 30

a)

0.2896

b)

-0.2543

c)

0.1152

d)

-0.1150

30.

a)

+/- 1.1691

b)

+/- 1.8965

c)

+/- 1.9901

d)

+/1 1.2315

31.

Solve using the properties log₅x - log₅2 = log₅15

a)

6

b)

8

c)

15

d)

30

32.

Solve using the properties. 3log₄a = log₄27

a)

16

b)

27

c)

3

d)

9

33.

Solve using the properties.

log₂(4x) + log₂5 = log₂40

a)

2

b)

1

c)

10

d)

8

34.

Solve using the properties.

log₅7 + 1/2log₅4 = log₅x

a)

14

b)

7

c)

7/4

d)

2

35.

Solve for x: log₆x + log₆(x-5) = 2

a)

√7

b)

9

c)

4

d)

none of these

36.

Which logarithmic equation is equivalent to $e^x=8$ ?

a)

$e^{\ln\left(x\right)}=8$

b)

$\ln\left(e^8\right)=x$

c)

$\ln x=8$

d)

$\ln8=x$

37.

Which logarithmic equation is equivalent to $e^x=4$ ?

a)

$\ln4=x$

b)

$\ln x=4$

c)

$e^{\ln\left(4\right)}=x$

d)

$\ln\left(e^x\right)=4$

38.

Write the equation $4^3=64$ in logarithmic form.

a)

$\log_43=\ 64$

b)

$\log_34=\ 64$

c)

$\log_{64}4=\ 3$

d)

$\log_464=\ 3$

39.

Solve for x

e^{\left(3x-1\right)}=2

a)

0.564

b)

0

c)

0.693

d)

1.079

40.

Solve for x

2\ln x=0.4

a)

1.221

b)

0.746

c)

1.492

d)

2.226

41.

Logarithmic functions are the inverse of...

a)

Linear Functions

b)

Exponential Functions

c)

Quadratic Functions

d)

Polynomial Functions

42.

The common logarithm has what base?

a)

10

b)

0

c)

e

d)

-e

43.

log 10 =

a)

1

b)

2

c)

3

d)

4

44.

log 100 =

a)

1

b)

2

c)

3

d)

4

45.

log 0.1 =

a)

-2

b)

-1

c)

1

d)

2

46.

log (1/10) =

a)

-2

b)

-1

c)

1

d)

2

47.

log 1000 - log 0.01 =

a)

2

b)

3

c)

4

d)

5

48.

log 100³ =

a)

3

b)

4

c)

5

d)

6

49.

log 1000 + log 0.1 - log 0.01 =

a)

0

b)

2

c)

3

d)

4

50.

log 100√10

a)

1/2

b)

3/2

c)

5/2

d)

2

51.

Expand: $\log_6\left(\frac{5}{4y}\right)$

a)

$\log_65\ +\ \log_64\ +\ \log_6y$

b)

$\log_65\ -\ \log_64\ +\ \log_6y$

c)

$\log_65\ -\ \log_64\ -\ \log_6y$

d)

$\log_65\ -\ \log_64y$

52.

Condense: $\log\left(5x+2\right)\ -\ \log3\ -\ \log x$

a)

$\log\left(\frac{3x}{5x\ +\ 2}\right)$

b)

$\log\left(\frac{5x\ +\ 2}{3x}\right)$

c)

$\log\left(3x\left(5x+2\right)\right)$

d)

$\log\left(\frac{5x\ +\ 2}{3\ -\ x}\right)$

53.

Expand: $\log_4\left(3x^2\right)$

a)

$2\log_43x$

b)

$2\log_43\ +\ 2\log_4x$

c)

$\log_43\ +\ 2\log_4x$

d)

$2\log_43\ +\ \log_4x$

54.

Condense: $\frac{1}{2}\log_3x\ +\ 5\log_3y$

a)

$\log_3\left(\sqrt{xy^5}\right)$

b)

$\log_3\left(\sqrt{x}y^5\right)$

c)

$\log_3\left(5\sqrt{x}y\right)$

d)

$\log_3\left(xy\right)^{\frac{5}{2}}$

55.

Evaluate using Change of Base Formula: $\log_45$ Round to the nearest hundredth.

(a)

56.

Expand:

a)

$\frac{1}{3}\left(2\log_5x\ +\ 3\log_5y\ -\ 5\log_5z\right)$

b)

$\frac{1}{3}\left(2\log_5x\ -\ 3\log_5y\ -\ 5\log_5z\right)$

c)

$\frac{1}{3}\left(2\log_5x\ +\ 3\log_5y\ +\ 5\log_5z\right)$

d)

$2\log_5x\ +\ 3\log_5y\ -\ 5\log_5z$

57.

Condense: $3\ln x\ +\ 8\ln y$

a)

$\ln\left(\frac{x^3}{y^8}\right)$

b)

$\ln\left(x^3y^8\right)$

c)

$24\ln\left(xy\right)$

d)

$11\ln\left(xy\right)$

58.

Condense: $3\log_bx\ -\ \log_by\ -\ 5\log_bz$

a)

$\log_b\ \frac{x}{yz^5}$

b)

$\log_b\ \frac{x^3}{yz^{ }}$

c)

$\log\frac{x}{yz^5}$

d)

$\log_b\ \frac{x^3}{yz^5}$

59.

Evaluate using Change of Base Formula: $\log_94$ Round to the nearest hundredth.

(a)

60.

Evaluate using properties of logarithms (no calculator): $3\ln e\ \ +\ 2\ln1$

(a)

61.

Evaluate using properties of logarithms (no calculator): $\log_64\ +\ \log_654$

(a)

62.

Evaluate using properties of logarithms (no calculator): $\log_5100\ -\ 2\log_52$

(a)

63.

Evaluate using properties of logarithms (no calculator): $\log2\ +\ \log10\ +\ \log5$

(a)

64.

Condense the Logarithm $5\log_{ }a\ -\ 25\log_{ }b$

a)

$\log\ \left(a^5+b^{25}\right)$

b)

$\log\ \left(a^5-b^{25}\right)$

c)

$\log\ \left(ab\right)^{25}$

d)

$\log_{ }\ \left(\frac{a^5}{b^{25}}\right)$

65.

Condense $3\log_{ }x+4\log_{ }y\ +\log_{ }z$

a)

$\log_{ }\ x^3y^4z$

b)

$12\log_{ }\ xyz$

c)

$\log_{ }\ 3x4yz$

d)

logx³y³z³

66.

Use the change-of-base formula to evaluate $\log_211$

a)

$3.459$

b)

$4.359$

c)

$5.123$

d)

2.345

67.

Simplify: $\log_4\left(x+4\right)-\log_4\left(x-5\right)$

a)

$\log_49$

b)

$\log_4\left(2x-1\right)$

c)

$\log_4\left(x^2-x-20\right)$

d)

$\log_4\left(\frac{x+4}{x-5}\right)$

68.

Simplify: $2\log_3\left(11x\right)$

a)

$\log_3\left(22x\right)$

b)

$\log_3\left(121x\right)$

c)

$\log_3\left(121x^2\right)$

d)

$\log_3\left(11x^2\right)$

69.

Simplify: $\frac{1}{4}\log_516+3\log_5x$

a)

$\log_5\left(4x^3\right)$

b)

$\log_5\left(2x^3\right)$

c)

$\log_5\left(6x\right)$

d)

$\log_5\left(\frac{2}{x^3}\right)$

70.

Simplify: $\frac{1}{4}\log_281+\frac{1}{2}\log_249$

a)

$\log_221$

b)

$\log_210$

c)

$\log_2\left(\frac{3}{7}\right)$

d)

$\log_244.75$

71.

Simplify: $\frac{1}{2}\log_964+\log_9x$

a)

$\log_9\left(32x\right)$

b)

$\log_9\left(8x\right)$

c)

$\log_9\left(\frac{8}{x}\right)$

d)

$\log_98x$

72.

Write the expression as a single logarithm. Then simplify if possible.
log 6 - log 3 + 2 log 7

a)

log 98

b)

log 78

c)

log 56

d)

log 45

73.

Expand using the properties of Logaritms $\log_{ }\ \frac{x^3}{y^4z}$

a)

$\log_{ }x+4\log_{ }y\ +\log_{ }z$

b)

$3\log_{ }x-4\log_{ }y\ -\log_{ }z$

c)

$3\log_{ }x+4\log_{ }y\ +\log_{ }z$

d)

$3\log_{ }x-4\log_{ }y\ +\log_{ }z$

74.

Use these and other properties of logarithms to evaluate the expression.

$\log_232\ -\ 6^{\log_63}$

a)

$2$

b)

$-2$

c)

$8$

d)

$3$

75.

Expand the logarithm.
$\log_4\sqrt{x^3y}$

a)

$\frac{1}{2}\log_4\left(x\right)-\frac{1}{2}\log_4\left(y\right)$

b)

$\frac{3}{2}\log_4\left(x\right)-\frac{1}{2}\log_4\left(y\right)$

c)

$\frac{1}{2}\log_4\left(x\right)-\log_4\left(y\right)$

d)

$\frac{3}{2}\log_4\left(x\right)-\log_4\left(y\right)$

76.

Use the change-of-base formula to evaluate $\log_7\ \frac{3}{16}$ rounded to two decimal places

a)

$0.82$

b)

$0.86$

c)

$0.85$

d)

$0.87$

77.

Expand the logarithm.
$\log\frac{x}{y^6}$

a)

$\log x+6\log y$

b)

$\log x-6\log y$

c)

$\log x+\log6y$

d)

$\log x-\log6y$

78.

Expand . $\log_6\left(\frac{5x^3}{y}\right)$

a)

$\log_65x^3-\log_6y$

b)

$\log_65+\log_6x^3-\log_6y$

c)

$\log_65+3\log_6x-\log_6y$

d)

$\log_65+3\log_6x+\log_6y$

79.

Evaluate $\log_52+\log_520-\log_54$ . Use the table to approximate the value of the logarithmic expression or use the change-of-base formula to simplify.

a)

$1.4307$

b)

$-1.4307$

c)

$1.3470$

d)

$-1.347$

80.

Evaluate $\log_6\ \frac{1}{36}+\log_636-5^{\log_61}$ . Use the table to approximate the value of the logarithmic expression or use the change-of-base formula to simplify.

a)

$1$

b)

$5$

c)

$12$

d)

$0$

81.

Evaluate $\log_6\ \frac{1}{216}+\log_24+\log_2\ \frac{1}{8}$ . Use the table to approximate the value of the logarithmic expression or use the change-of-base formula to simplify.

a)

$4$

b)

$-4$

c)

$2$

d)

$5$

82.

Evaluate $\log_330-\log_4\ 4^5$ . Use the table to approximate the value of the logarithmic expression or use the change-of-base formula to simplify.

a)

$1.9041$

b)

$-1.9041$

c)

$19,041$

d)

$9,041$

83.

Evaluate $1000^{\log_{10}4}-\log_464+25^{\log_510}$ . Use the table to approximate the value of the logarithmic expression or use the change-of-base formula to simplify.

a)

$161$

b)

$61$

c)

$-161$

d)

$126$

Logarithms

Total questions: 83

Worksheet time: 4hrs 52mins

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