SOLVING LOGARITHMIC EQUATIONS

Solving Logarithmic Equations : 

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

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Solving Logarithmic Equations - Example

Problem 1 :

Solve for x :

log₂x  =  1/2

Solution :

log₂x  =  1/2

Convert to exponential form. 

x  =  2^1/2

x  =  √2

Problem 2 :

Solve for x :

log_1/5x  =  3

Solution :

log_1/5x  =  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 :

log_x125√5  =  7

Solution :

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

Problem 5 :

Solve for x :

log_x0.001  =  -3

Solution :

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

Problem 6 :

Solve for x : 

x + 2log₂₇9  =  0

Solution :

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

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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