PROPERTIES OF LOGARITHMS WORKSHEET

Properties of Logarithms Worksheet : 

Worksheet given in this section will be much useful for the students who would like to practice solving problems on properties of logarithms. 

Before look at the worksheet, if you would like to learn the properties of logarithms, 

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Properties of Logarithms Worksheet - Problems

Problem 1 :

Let b > 0 and b ≠ 1. Express y = b^x in logarithmic form. Also state the domain and range of the logarithmic function.

Problem 2 :

Compute : 

log₉ 27 − log₂₇ 9

Problem 3 :

Solve for x :

log₈x + log₄x + log₂x  =  11

Problem 4 :

Solve for x : 

log₄2^8x  =  2^log₂⁸

Properties of Logarithms Worksheet - Solutions

Problem 1 :

Let b > 0 and b ≠ 1. Express y = b^x in logarithmic form. Also state the domain and range of the logarithmic function.

Solution :

y  =  b^x

log_by  =  x

Domain  =  (0, ∞)

Range  =  (-∞, ∞)

Problem 2 :

Compute : 

log₉ 27 − log₂₇ 9

Solution :

Therefore,

log₉27 − log₂₇9  =  3/2 - 2/3

log₉27 − log₂₇9  =  9/6 - 4/6

log₉27 − log₂₇9  =  (9 - 4) / 6

log₉27 − log₂₇9  =  5/6

Problem 3 :

Solve for x :

log₈x + log₄x + log₂x  =  11

Solution :

log₈x + log₄x + log₂x  =  11

(1 / log_x8) + (1 / log_x4) + (1 / log_x2)  =  11

(1 / log_x2³) + (1 / log_x2²) + (1 / log_x2)  =  11

(1 / 3log_x2)  + (1 / 2log_x2) +  (1 / log_x2)  =  11

(2 / 6log_x2)  + (3 / 6log_x2) +  (6 / 6log_x2)  =  11

(2 + 3 + 6) / 6log_x2  =  11

 11 / 6log_x2  =  11

Take reciprocal on both sides. 

6log_x2 / 11  =  1 / 11

Multiply each side by 11. 

6log_x2  =  1

Divide each side by 6. 

log_x2  =  1/6

Change to exponential form. 

2  =  x^1/6

Take power 6 on both sides. 

2⁶  =  (x^1/6)⁶

64  =  x

Therefore, the value of 'x' is 64.

Problem 4 :

Solve for x : 

log₄2^8x  =  2^log₂⁸

Solution : 

log₄2^8x  =  2^log₂⁸ -----(1)

Find the value of log₂8.

log₂8  =  log₂2³

log₂8  =  3log₂2

log₂8  =  3(1)

log₂8  =  3

Substitute the value of log₂8 in (1). 

(1)-----> log₄2^8x  =  2³

(1)-----> log₄2^8x  =  8

Change to exponential form. 

2^8x  =  4⁸

2^8x  =  (2²)⁸

2^8x  =  2¹⁶

Equate the exponents. 

8x  =  16

Divide each side by 8.

x  =  2

Therefore, the value of 'x' is 2. 

After having gone through the stuff given above, we hope that the students would have understood how to solve problems using the properties of logarithms. 

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