# 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 :

logba  =  logx⋅ logbx

logba  =  logxa / logxb

Example 1 :

Solve the following equation :

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

Solution :

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

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

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

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

2x  =  14

Divide each side by 2.

x  =  7

So, 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)

Subtract log5 √(x + 4) from each side.

log3 (5x - 2) - log3 √(x + 4) - 1/2  =  0

log3 (5x - 2) - log3 √(x + 4)  =  1/2

Use quotient rule.

log3[√(5x - 2) / √(x + 4)]  =  1/2

Convert to exponential form.

[√(5x - 2) / √(x + 4)]  =  31/2

(5x - 2) / (x + 4)  =  √3

Square each side.

(5x - 2) / (x + 4)  =  3

Multiply each side by (x + 4).

5x - 2  =  3 (x + 4)

5x - 2  =  3x + 12

Subtract 3x from each side.

2x - 2  =  12

2x  =  14

Divide each side by 2.

x  =  7

So, the value of x is 7. After having gone through the stuff given above, we hope that the students would have understood how to solve logarithmic equations with radicals.

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