EVALUATING LOGARITHMS WITHOUT A CALCULATOR WORKSHEET

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Find :

1) log₃9

2) log₂32

3) log₂√2

4) log₄√2

5) log₃3√3

6) log₆1

7) log₈8

8) log₈(1/8)

9) log_1/8(1/8)

10) log_√2(1/√2)

11) log₂(1/√2)

12) log₈(1/√2)

Solution

Problem 13 :

Solve ln (4x − 7) = ln (x + 5)

Solution

Problem 14 :

log₂(5x − 17) = 3

Solution

Problem 15 :

log₁₀(9x + 1) = 3

Solution

Problem 16 :

log₈√(1 - x) = 1/3

Solution

Problem 17 :

Solve the simultaneous equations:

log₃ x + log₅ y = 6

log₃ x - log₅ y = 2

Solution

Problem 18 :

Solve the equation

3(1 + log x) = 6 + log x

Solution

Problem 19 :

Solve log₃ [(2 + x) / (2 - x)] = 3

Solution

Answer Key

1) 2

2) 5

3) 1/2

4) 1/4

5) 3/2

6) 0

7) 1

8) -1

9) 1

10) -1

11) -1/2

12) -1/6

13) x = 4

14) x = 5

15) x = 111

16) x = -3

17) x = 9 and y = 625

18) x = 31.6

19) x = 56/3

Without using a calculator, simplify.

1) log 8 / log 2

2) log 9 / log 3

3) log 4 / log 8

4) log 5 / log (1/5)

5) log (0.5) / log 2

6) log 8 / log (0.25)

7) log 2^b / log 8

8) log 4 / log 2^a

Solution

Problem 9 :

Under certain conditions, the wind speed s (in knots) at an altitude of h meters above a grassy plain can be modeled by the function

s(h) = 2 ln 100h

a. By what amount does the wind speed increase when the altitude doubles?

b. Show that the given function can be written in terms of common logarithms as

s(h) = (2 /log e) (log h + 2)

Solution

Problem 10 :

-3 ln x = -24

Solution

Problem 11 :

4 - 3log (5x) = 16

Solution

Problem 12 :

log₃(x - 1) = -2

Solution

Problem 13 :

log₂(x² - 4) = log₂21

Solution

Problem 14 :

The function

s(d) = 0.159 + 0.118 log d

relates the slope, s, of a beach to the average diameter, d, in millimeters, of the sand particles on the beach. Which beach has a steeper slope: beach A, which has d = 0.0625, or beach B, which has very coarse sand with d = 1 ? Justify your decision.

Solution

Problem 15 :

The function

S(d) = 93 log d + 65

relates the speed of the wind, S, in miles per hour, near the center of a tornado to the distance that the tornado travels, d, in miles.

a) If a tornado travels a distance of about 50 miles, estimate its wind speed near its center.

b) If a tornado has sustained winds of approximately 250 mph, estimate the distance it can travel.

Solution

Answer Key

1) 3

2) 2

3) 2/3

4) -1

5) 1

6) -3/2

7) b/3

8) 2/a

9) a) the wind speed will increase by 1.38 knots.

b) (log h² + 4) / log e

10) x = e⁸

11) x = 1/50000

12) x = 10/9

13) x = -5 and 5

14) Slope for Beach B > Slope for Beach A

15) a) 222.91

b) Approximately 95 miles

Find the value of y.

1) log₅ 25 = y

2) log₃1 = y

3) log₁₆ 4 = y

4) log₂ (1/8) = y

5) log₅1 = y

6) log₂ 8 = y

7) log₇ (1/7) = y

8) log₃ (1/9) = y

9) log_y 32 = 5

10) log₉ y = -1/2

11) log₄ (1/8) = y

12) log₉ (1/81) = y

Solution

Problem 13 :

Describe the similarities and difference between in solving the equations

4^{5x - 2} = 16 and log₄(10x + 6) = 1

Then solve the each equation

Solution

Problem 14 :

For a sound with intensity I (in watts per square meter) the loudness L(I) of the sound (in decibels) is given by the function

L(I) = 10 log (I/I₀)

Where I0 is the intensity of barely audible sound (about 10^-12 watts per square meter) An artist in a recording studio turns up the volume of a track so that the intensity of the sound doubles. By how many decibels does the loudness increase ?

Solution

Problem 15 :

The length ℓ (in centimeters) of a scalloped hammerhead shark can be modeled by the function

ℓ = 266 − 219e^−0.05t

where t is the age (in years) of the shark. How old is a shark that is 175 centimeters long?

Solution