# SOLVING LOGARITHMIC EQUATIONS

Solving Logarithmic Equations :

In this section, you will learn how to solve logarithmic equations.

To know about logarithm in detail,

## Solving Logarithmic Equations - Example

Problem 1 :

Solve for x :

log2x  =  1/2

Solution :

log2x  =  1/2

Convert to exponential form.

x  =  21/2

x  =  √2

Problem 2 :

Solve for x :

log1/5x  =  3

Solution :

log1/5x  =  3

Convert to exponential form.

x  =  (1/5)3

x  =  13/53

x  =  1/125

Problem 3 :

Solve for y :

log3y  =  -2

Solution :

log3y  =  -2

Convert to exponential form.

y  =  3-2

y  =  1/32

y  =  1/9

Problem 4 :

Solve for x :

logx125√5  =  7

Solution :

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

Problem 5 :

Solve for x :

logx0.001  =  -3

Solution :

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

Problem 6 :

Solve for x :

x + 2log279  =  0

Solution :

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

Problem 7 :

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

Solution :

2logx  =  4log3

Divide each side by 2.

logx  =  (4log3) / 2

logx  =  2log3

logx  =  log32

logx  =  log9

x  =  9

Problem 8 :

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

Solution :

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 After having gone through the stuff given above, we hope that the students would have understood how to solve logarithmic equations.

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