SOLVING LOGARITHMIC EQUATIONS WORKSHEET

Solve for x :

1) log₂x  =  1/2

2) log_1/5x  =  3

3) log_x125√5  =  7

4) log_x0.001  =  -3

5) log₅(5log₃x)  =  2

6) x + 2log₂₇9  =  0

7) If 2logx  =  4log3,  then find the value of x. 

8) If 3x is equal to log(0.3) to the base 9, then find the value of x.  

Solve for x :

9) log₅ √(7x - 4) - 1/2  =  log₅ √(x + 2)

Solve for x :

10) log₃x + log₉x + log₈₁x  =  7/4

Detailed Answer Key

Answer (1) : 

log₂x  =  1/2

Convert to exponential form. 

x  =  2^1/2

x  =  √2

Answer (2) :

log_1/5x  =  3

Convert to exponential form. 

x  =  (1/5)³

x  =  1³/5³

x  =  1/125

Answer (3) :

log_x125√5  =  7

Convert to exponential form. 

125√5  =  x⁷

 5 ⋅ 5 ⋅ 5 ⋅ √5  =  x⁷

Each 5 can be expressed as (√5 ⋅ √5).

Then,

√5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5  =  x⁷

√5⁷  =  x⁷

Because the exponents are equal, bases can be equated. 

x  =  √5

Answer (4) :

log_x0.001  =  -3

Convert to exponential form. 

0.001  =  x^-3

1/1000  =  1/x³

Take reciprocal on both sides. 

1000  =  x³

10³  =  x³

Because the exponents are equal, bases can be equated. 

10  =  x

Answer (5) :

log₅(5log₃x)  =  2

Convert to exponential form. 

5log₃x  =  5²

5log₃x  =  25

Divide each side by 5.

log₃x  =  5

Convert to exponential form. 

x  =  3⁵

x  =  243

Answer (6) :

x + 2log₂₇9  =  0

x  =  -2log₂₇9

x  =  log₂₇9^-2

Convert to exponential form. 

27^x  =  9^-2

(3³)^x  =  (3²)^-2

3^3x  =   3^-4

Because the bases are equal, exponents can be equated. 

3x  =  -4

x  =  -4/3

Answer (7) : 

2logx  =  4log3

Divide each side by 2.

logx  =  (4log3) / 2

logx  =  2log3

logx  =  log3²

logx  =  log9

x  =  9

Answer (8) : 

From the information given, we have

3x  =  log₉(0.3)

Solve for x.

3x  =  log₉(1/3)

3x  =  log₉1 - log₉3

3x  =  0 - log₉3

3x  =  - log₉3

3x  =  - 1 / log₃9

3x  =  - 1 / log₃3²

3x  =  - 1 / 2log₃3

3x  =  - 1 / 2(1)

3x  =  -1/2

x  =  -1/6

Answer (9) :

log₅ √(7x - 4) - 1/2  =  log₅ √(x + 2)

Subtract log₅ √(x + 2) from each side. 

log₅ √(7x - 4) - log₅ √(x + 2) - 1/2  =  0

Add 1/2 to each side. 

log₅ √(7x - 4) - log₅ √(x + 2)  =  1/2

Use quotient rule.

log₅[√(7x - 4) / √(x + 2)]  =  1/2

Convert to exponential form.

[√(7x - 4) / √(x + 2)]  =  5^1/2 

(7x - 4) / (x + 2)  =  √5 

Square each side.  

(7x - 4) / (x + 2)  =  5 

Multiply each side by (x + 2). 

7x - 4  =  5 (x + 2)

7x - 4  =  5x + 10

Subtract 5x from each side.

2x - 4  =  10

Add 4 to each side.

2x  =  14

Divide each side by 2.

x  =  7

Answer (10) :

log₃x + log₉x + log₈₁x  =  7/4

(1 / log_x3)  +  (1 / log_x9)  +  (1 / log_x81)  =  7/4

(1 / log_x3)  +  (1 / log_x9)  +  (1 / log_x81)  =  7/4

(1 / log_x3)  +  (1 / log_x3²)  +  (1 / log_x3⁴)  =  7/4

(1 / log_x3)  +  (1 / 2log_x3)  +  (1 / 4log_x3)  =  7/4

(4 / 4log_x3)  +  (2 / 4log_x3)  +  (1 / 4log_x3)  =  7/4

(4 + 2 + 1) / 4log_x3  =  7/4

7 / 4log_x3  =  7/4

Multiply each side by 4/7.

1 / log_x3  =  1

log₃x  =  1

Convert to exponential form. 

x  =  3¹

x  =  3

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