SOLVING LOGARITHMIC EQUATIONS WORKSHEET

Solve for x :

1) log2x  =  1/2

2) log1/5x  =  3

3) logx125√5  =  7

4) logx0.001  =  -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  =  21/2

x  =  √2

Answer (2) :

log1/5x  =  3

Convert to exponential form. 

x  =  (1/5)3

x  =  13/53

x  =  1/125

Answer (3) :

logx125√5  =  7

Convert to exponential form. 

125√5  =  x7

 5 ⋅ 5 ⋅ 5 ⋅ √5  =  x7

Each 5 can be expressed as (⋅ 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) :

logx0.001  =  -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. 

27x  =  9-2

(33)x  =  (32)-2

33x  =   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)]  =  51/2 

(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 / logx3)  +  (1 / logx9)  +  (1 / logx81)  =  7/4

(1 / logx3)  +  (1 / logx9)  +  (1 / logx81)  =  7/4

(1 / logx3)  +  (1 / logx32)  +  (1 / logx34)  =  7/4

(1 / logx3)  +  (1 / 2logx3)  +  (1 / 4logx3)  =  7/4

(4 / 4logx3)  +  (2 / 4logx3)  +  (1 / 4logx3)  =  7/4

(4 + 2 + 1) / 4logx3  =  7/4

7 / 4logx3  =  7/4

Multiply each side by 4/7.

1 / logx3  =  1

log3x  =  1

Convert to exponential form. 

x  =  31

x  =  3

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