SOLVING LOGARITHMIC EQUATIONS WITH RADICALS

About "Solving logarithmic equations with radicals"

Solving logarithmic equations with radicals 

Here we are going to see how to solve logarithmic equations with radicals.

Basic rules in logarithm

Product rule :

log a (m ⋅ n)  =  logam + logan

Quotient rule :

log a (m / n)  =  logam - logan

Power rule :

log a mn  =  n logam

Change of base rule :

log a m  =  logb⋅ logab

log a a  =  1

Let us look into some examples to understand the concept of solving logarithmic equations with radicals.

Example 1 :

Solve the following equation

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

Solution :

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

In order to combine both logarithmic terms, we have to subtract both sides by log5 √(x + 2) and add both sides by 1/2.

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

log5[(7x - 4) / √(x + 2)]  =  1/2

[(7x - 4) / √(x + 2)]  =  51/2 

[(7x - 4) /(x + 2)]  =  5 

Taking square on both sides 

(7x - 4) /(x + 2)  =  5 

Multiply both sides by (x + 2) 

7x - 4  =  5 (x + 2)

7x - 4  =  5x + 10

Subtract both sides by 5x

7x - 5x - 4  =  10

2x - 4  =  10

Add both sides by 4

2x  =  10 + 4

2x  =  14

Divide both sides by 2

x  =  14/2  =  7

Hence the value of x is 7.

Example 2 :

Solve the following equation

log3 √(5x - 2) - (1/2)  =  log3 √(x + 4)

Solution :

log3 √(5x - 2) - (1/2)  =  log3 √(x + 4)

In order to combine both logarithmic terms, we have to subtract both sides by log3 √(x + 4) and add both sides by 1/2.

log3 (5x - 2) - log3 √(x + 4)  =  (1/2)

log3[(5x - 2) / √(x + 4)]  =  1/2

[√(5x - 2) / √(x + 4)]  =  31/2 

[√(5x - 2) /(x + 4)]  =  √3 

Taking square on both sides 

(5x - 2) /(x + 4)  =  3 

Multiply both sides by (x + 4) 

5x - 2  =  3 (x + 4)

5x - 2  =  3x + 12

Subtract both sides by 3x

5x - 3x - 2  =  12

2x - 2  =  12

Add both sides by 2

2x  =  12 + 2

2x  =  14

Divide both sides by 2

x  =  14/2  =  7

Hence the value of x is 7. 

After having gone through the stuff given above, we hope that the students would have understood "Solving logarithmic equations with radicals". 

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