PROPERTIES OF LOGARITHMS WORKSHEET

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

Detailed Answer Key

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. 

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