**Solving logarithmic equations with radicals **

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

**Product rule :**

log _{a} (m ⋅ n) = log_{a}m + log_{a}n

**Quotient rule :**

log _{a} (m / n) = log_{a}m - log_{a}n

**Power rule :**

log _{a} m^{n} = n log_{a}m

**Change of base rule :**

log _{a} m = log_{b}m ⋅ log_{a}b

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

log_{5} √(7x - 4) - (1/2) = log_{5} √(x + 2)

**Solution :**

log_{5} √(7x - 4) - (1/2) = log_{5} √(x + 2)

In order to combine both logarithmic terms, we have to subtract both sides by log_{5} √(x + 2) and add both sides by 1/2.

log_{5} √(7x - 4) - log_{5} √(x + 2) = 1/2

log_{5}[√(7x - 4) / √(x + 2)] = 1/2

[√(7x - 4) / √(x + 2)] = 5^{1/2 }

[√(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

log_{3} √(5x - 2) - (1/2) = log_{3} √(x + 4)

**Solution :**

log_{3} √(5x - 2) - (1/2) = log_{3} √(x + 4)

In order to combine both logarithmic terms, we have to subtract both sides by log_{3} √(x + 4) and add both sides by 1/2.

log_{3} √(5x - 2) - log_{3} √(x + 4) = (1/2)

log_{3}[√(5x - 2) / √(x + 4)] = 1/2

[√(5x - 2) / √(x + 4)] = 3^{1/2 }

[√(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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