**Solving logarithmic equations with variables :**

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

**Step 1 :**

To solve logarithmic equations, first we have to convert the logarithmic form to exponential form.

The picture given below will illustrate the first step.

**Step 2 :**

Now express the number in the left side in exponential form.

**Step 3 :**

If the bases are equal, then powers will be equal.

**Step 4 :**

From step 3, we get the value of unknown.

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

**Example 1 :**

Solve the following equation

log_{5} x = -3

**Solution :**

**Given logarithmic form :**

log_{5} x = -3

**Exponential form :**

x = 5^{-3}

x = 1/5^{3}

x = 1/125

Hence the value of x is 1/125.

**Example 2 :**

Solve the following equation

x = log_{1/4} 64

**Solution :**

**Given logarithmic form :**

x = log_{1/4} 64

**Exponential form :**

(1/4)^{x} = 64

(4^{-1})^{x} = 4^{3}

4^{-x} = 4^{3}

-x = 3

Hence the value of x is -3.

**Example 3 :**

Solve the following equation

log_{x} 8 = 2

**Solution :**

**Given logarithmic form :**

log_{x} 8 = 2

**Exponential form :**

8 = x^{2}

√8 = x

x = 2√2

Hence the value of x is 2√2.

**Example 4 :**

Solve the following equation

log_{2} x = 1/2

**Solution :**

**Given logarithmic form :**

log_{2} x = 1/2

**Exponential form :**

x = 2^{1/2}

x = √2

Hence the value of x is √2.

**Example 5 :**

Solve the following equation

log_{1/5} x = 3

**Solution :**

**Given logarithmic form :**

log_{1/5} x = 3

**Exponential form :**

x = (1/5)^{3}

x = 1^{3}/5^{3}

x = 1/125

Hence the value of x is 1/125.

**Example 6 :**

Solve the following equation

log_{3} y = -2

**Solution :**

**Given logarithmic form :**

log_{3} y = -2

**Exponential form :**

y = 3^{-2}

y = 1/3^{2}

y = 1/9

Hence the value of y is 1/9.

**Example 7 :**

Solve the following equation

log_{x} 125 √5 = 7

**Solution :**

**Given logarithmic form :**

log_{x} 125 √5 = 7

**Exponential form :**

125 √5 = x^{7}

5 ⋅ 5 ⋅ 5 ⋅ √5 = x^{7}

Every 5 can be expressed as the product of √5 ⋅ √5

√5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 = x^{7}

√5^{7 }= x^{7}

Since the powers are equal, the value of bases will be equal.

x = √5

Hence the value of x is √5.

**Example 8 :**

Solve the following equation

log_{x} 0.001 = -3

**Solution :**

**Given logarithmic form :**

log_{x} 0.001 = -3

**Exponential form :**

0.001 = x^{-3}

1/1000 = x^{-3}

(1/10)^{3} = x^{-3}

(10^{-1})^{3} = x^{-3}

10^{-3} = x^{-3}

Hence the value of x is 10.

**Example 9 :**

Solve the following equation

x + 2 log_{27} 9 = 0

**Solution :**

x + 2 log_{27} 9 = 0

x = - 2 log_{27} 9

x = log_{27} 9^{-2}

x = log_{27} 9^{-2}

27^{x} = 9^{-2}

Express 27 and 9 as the exponent of 3.

(3^{3})^{x }= 9^{-2 }==> 3^{3x} = (3^{2})^{-2}

3^{3x} = 3^{-4}

3x = -4

x = -4/3

Hence the value of x is -4/3.

**Example 10 :**

Solve the following equation

log_{x} 7^{1/6} = 1/3

**Solution :**

log_{x} 7^{1/6} = 1/3

7^{1/6} = x^{1/3}

7^{1/(2}^{⋅}^{3)} = x^{1/3}

(7^{1/2})^{(1/3)} = x^{1/3}

(√7)^{(1/3)} = x^{1/3}

x = √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 variables".

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